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• Probability For Class 10

Probability Class 10 Notes Chapter 15

According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 14.

CBSE Class 10 Maths Probability Notes:- Download PDF Here

Class 10 maths chapter 15 probability notes.

CBSE Class 10 Maths Chapter 15 Probability Notes are provided here in detail. In this article, we are going to learn the definition of probability, experimental probability, theoretical probability and the different terminologies used in probability with complete explanations.

Students can refer to the short notes and MCQ questions along with separate solution pdf of this chapter for quick revision from the links below:

• Probability Short Notes
• Probability MCQ Practice Questions
• Probability MCQ Practice Solutions

What Is Probability?

The branch of mathematics that measures the uncertainty of the occurrence of an event using numbers is called probability. The chance that an event will or will not occur is expressed on a scale ranging from 0-1. It can also be represented as a percentage, where 0% denotes an impossible event and 100 % implies a certain event. Probability of an Event E is represented by P(E). For example, the probability of getting a head when a coin is tossed is equal to 1/2. Similarly, the probability of getting a tail when a coin is tossed is also equal to 1/2. Hence, the total probability will be: P(E) = 1/2 + 1/2 = 1 Know more about probability by clicking here .

Event and outcome

An Outcome is a result of a random experiment. For example, when we roll a dice getting six is an outcome. An Event  is a set of outcomes. For example, when we roll dice, the probability of getting a number less than five is an event. Note: An event can have a single outcome.

To know more about Types of Events, visit here .

Experimental Probability

Experimental probability can be applied to any event associated with an experiment that is repeated a large number of times. A trial is when the experiment is performed once. It is also known as empirical probability . Experimental or empirical probability: P(E) =Number of trials  where the event occurred/Total Number of Trials Example: In a day, a shopkeeper is able to sell 15 balls, out of which 6 were red balls. Find the probability of selling red balls on the next day of his sales. Given, the total number of balls sold = 15 Number of red balls sold = 6 Probability of red balls = 6/15 = 2/5

To know more about Experimental Probability, visit here .

Theoretical Probability

Theoretical Probability, P(E) = Number of Outcomes Favourable to E / Number of all possible outcomes of the experiment

Here we assume that the outcomes of the experiment are equally likely . Example: Find the probability of picking up a red ball from a basket that contains 5 red and 7 blue balls. Solution: Number of possible outcomes = Total number of balls = 5+7 = 12 Number of favourable outcomes = Number of red balls = 5 Hence, Probability, P(red) = 5/12

For more information on Probability, watch the below video

To know more about Theoretical Probability, visit here .

Elementary Event

An event having only one outcome of the experiment is called an elementary event . Example: Take the experiment of tossing a coin n number of times. One trial of this experiment has two possible outcomes: Heads(H) or Tails(T). So for an individual toss, it has only one outcome, i.e. Heads or Tails.

Sum of Probabilities

The sum of the probabilities of all the elementary events of an experiment is one . Example: take the coin-tossing experiment. P(Heads) + P(Tails )

=  (1/2)+ (1/2) =1

Impossible Event

An event that has a 100% probability of occurrence is called a sure event . The probability of occurrence of a sure event is one . E.g., What is the probability that a number obtained after throwing a die is less than 7? So,  P(E) = P(Getting a number less than 7) = 6/6= 1

Range of Probability of an event

Probability can range between 0 and 1, where 0 probability means the event to be an impossible one and probability of 1 indicates a certain event i.e. 0 ≤P (E) ≤ 1.

Geometric Probability

Complementary Events

Complementary events are two outcomes of an event that are the only two possible outcomes. This is like flipping a coin and getting heads or tails.

The best example of complementary events is flipping a coin, where ‘getting a head’ complement the event of ‘getting a tail’. To know more about Complementary Events, visit here .

Probability for Class 10 Examples

A bag contains only lemon-flavoured candies. Arjun takes out one candy without looking into the bag. What is the probability that he takes out an orange-flavoured candy?

Let us take the number of candies in the bag to be 100.

Hence, the probability that he takes out an orange-flavoured candy is:

P (Taking orange-flavoured candy) = Number of orange-flavoured candies / Total number of candies.

= 0/100 = 0

Hence, the probability that Arjun takes out an orange-flavoured candy is 0.

This proves that the probability of an impossible event is 0.

A game of chance consists of spinning an arrow that comes to rest, pointing at any one of the numbers such as 1, 2, 3, 4, 5, 6, 7, or 8, and these are equally likely outcomes. What is the probability that it will point at? (i)8 (ii) Number greater than 2 (iii) Odd numbers

Sample Space = {1, 2, 3, 4, 5, 6, 7, 8}

Total Numbers = 8

(i) Probability that the arrow will point at 8:

Number of times we can get 8 = 1

P (Getting 8) = 1/8.

(ii) Probability that the arrow will point at a number greater than 2:

Number greater than 2 = 3, 4, 5, 6, 7, 8.

No. of numbers greater than 2 = 6

P (Getting numbers greater than 2) = 6/8 = 3/4.

(iii) Probability that the arrow will point at the odd numbers:

Odd number of outcomes = 1, 3, 5, 7

Number of odd numbers = 4.

P (Getting odd numbers) = 4/8 = ½.

Related Articles:

• NCERT Solutions for Class 10 Maths Chapter 15 Probability
• Class 10 Maths Chapter 15 Probability MCQs
• Important Questions Class 10 Maths Chapter 15 Probability

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• Solving Applied Probability Assignment Problems

Proven Techniques for Solving Applied Probability Assignment Problems

Understanding and solving problems related to probabilistic systems analysis and applied probability can be challenging yet rewarding. This guide aims to provide students with fundamental concepts and problem-solving techniques that can be applied to a variety of assignments in this area. By mastering these concepts, students can approach similar assignments with confidence and accuracy. The ability to work with probability operations such as complementation, union, and intersection, as well as applying probability axioms and principles, is crucial for tackling complex problems. This guide will cover essential methods for expressing and solve your probability assignment , interpreting results, and using visual tools like Venn diagrams to clarify relationships between events. With a solid grasp of these techniques, students can not only enhance their problem-solving skills but also gain a deeper understanding of how probability theory applies to real-world scenarios and academic challenges.

Basic Probability Concepts

Before diving into specific problems, it's essential to understand some foundational concepts of maths. Events and sample space are fundamental to probability problem in any math assignment . An event is a specific outcome or a set of outcomes of a random experiment, while the sample space is the set of all possible outcomes. Complementation, union, and intersection are crucial operations in probability. The complement of an event represents the set of outcomes not in the event. The union of events is the set of outcomes that are in at least one of the events, while the intersection of events is the set of outcomes that are in all the events.

Understanding Probability Operations

Understanding probability operations involves grasping how different events interact within a probability space. The primary operations include complementation, union, and intersection. Complementation refers to the set of outcomes not in a given event, while union combines all outcomes from multiple events, representing scenarios where at least one event occurs. Intersection, on the other hand, identifies outcomes where all specified events occur simultaneously. By mastering these operations, you can accurately express complex probabilistic scenarios, such as when at least one, none, or all events occur, and apply these concepts to solve various probability problems effectively.

At Least One Event Occurring

When considering the scenario where at least one of the events occurs, this means that at least one of the events A, B, or C happens. This can be represented using the union operation, which combines all possible outcomes where any of the events occur. The union of events A, B, and C is written as A ∪ B ∪ C. This union represents any outcome that falls within any of the three events.

At Most One Event Occurring

On the other hand, when considering at most one of the events occurring, it implies that no more than one of the events A, B, or C happens. This situation can be represented by the union of individual events and their complements, indicating that either none or only one of the events occurs. For example, this could be expressed as (A ∩ B') ∪ (A' ∩ B) ∪ (A' ∩ B' ∩ C'). Here, each term in the union represents a scenario where only one event happens or none at all.

None or All Events Occurring

None of the events occurring is represented by the intersection of the complements of the events. This means that if none of the events A, B, or C happen, then the outcome is in the complement of each event. This can be expressed as A' ∩ B' ∩ C'. Conversely, all three events occurring is represented by the intersection of the events, meaning all must happen simultaneously. This is written as A ∩ B ∩ C, indicating that the outcome is within all three events at the same time.

Drawing Venn Diagrams

Venn diagrams are a helpful tool to visualize these relationships. By drawing these diagrams, you can clearly see the intersections and unions of different events. This visualization can make complex relationships easier to understand. For example, when dealing with multiple events, using Venn diagrams can help you visualize how different sets overlap and interact, making it simpler to identify the relationships and solve related problems.

Using Probability Axioms

To prove certain probability relationships, you need to use the basic axioms of probability. For instance, to show that the probability of the intersection of two events is greater than or equal to the sum of their probabilities minus one, start by understanding the basic principles of probability and use the principle of inclusion and exclusion. This involves breaking down the problem step-by-step, using logical rules and axioms to arrive at the conclusion. This methodical approach helps ensure that your proof is solid and based on established probability principles.

Finding Complex Probabilities

When given specific conditions, use the properties of mutually exclusive events or independence to solve for complex probabilities. For example, to find the probability of a union of events under different conditions, break down the problem using the given conditions and apply the appropriate probability rules. This might involve using conditional probabilities or the law of total probability to account for various scenarios and calculate the desired probabilities accurately.

Card Problems

Problems involving drawing cards often require understanding combinations and the basic probability of events. For example, to find the probability that at least one card is an ace, you need to calculate the complementary event where neither card is an ace and subtract it from one. Similarly, finding the probability of both cards being of the same suit requires considering the total number of ways to draw two cards and the number of favorable outcomes. By understanding the structure of a deck of cards and using combinatorial methods, you can solve these types of problems systematically.

Random Number Selection

When dealing with random number selection problems, use geometric probability and area calculations for uniform distributions. Events like the magnitude of the difference of two numbers being greater than a certain value can be solved by setting up the appropriate integrals or geometric regions. For instance, if you are given a problem where two numbers are chosen at random within a certain range, you can visualize this as a geometric shape and calculate the probability based on the area of the relevant regions.

Special Dice Problems

For peculiar dice problems, determine the probability of outcomes based on the given conditions, such as the product of outcomes being proportional to their values. Use the total probability rule and properties of the dice to find the desired probabilities. For example, if you have a pair of dice with non-standard properties, you can calculate the probability of specific outcomes by considering the unique rules and constraints given in the problem.

By understanding and applying these fundamental concepts and techniques, students can effectively tackle a wide range of problems in probabilistic systems analysis and applied probability. Practice regularly, use visual aids like Venn diagrams, and apply the basic probability axioms to build a strong foundation in this subject. Whether working on assignments or preparing for exams, these skills will enhance your problem-solving abilities and deepen your understanding of probability.

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a) A face card [Ans 3/13]

a) A black ball [Ans 1/2]

b) A red or a black ball [Ans 11/14]

c) A red white or black ball [Ans 1]

d) Not a red ball [Ans 5/7]

e) Neither white nor black ball [Ans 2/7]

f) Not a white ball [Ans 11/14]

a) Difference of the numbers on the two dice is 2 [Ans 2/9]

b) Sum of numbers on the two dice is 10 [Ans 1/12]

c) Sum of numbers on two dice is more than 9 [Ans 1/6]

d) Six will come up at least once [Ans 11/36]

e) Both are odd number [Ans 1/4]

f) Six as a product. [Ans 1/9]

g) A doublet   [Ans 1/6]

h) Getting same number [Ans 1/6]

i) Getting different number [Ans 5/6]

j) A total of 9 or 11 [Ans 1/6]

a) A Leap Year [Ans 2/7]

b) In a non leap year [Ans 1/7]

a) Not red [Ans 5/6]

a)Two tails   [Ans 3/8]

a) Exactly one head  [Ans 2/4 = 1/2]

a) A heart [Ans 13/49]

a) A prime number [Ans 11/35]

a) A consonant [Ans 21/26]

a) Letter M or A

a) Will be a multiple of 5 [Ans 3/4]

a) Getting odd number

Answer:  a) 4/7   b) 2/7   c) 0   d) 5/7

a) Red or Black

a) No Head 

a) Probability of sure event…………

a) 1    b) 0      c)  0 ≤ P(E) ≤ 1    d)  0    e)  1

If the probability of a player winning a game is 0.79, then find the probability of his losing the same game.

Answer: 0.21

From the data 1, 4, 7, 9, 16, 21, 25, if all the even numbers are removed, then find the probability of getting a prime number.

Answer: 1/5

In a pack of playing cards one card is lost. From the remaining cards, a card is drawn at random. Find the probability that the drawn card is queen of heart, if the lost card is a black card.

Answer: 1/51

A carton consists of 60 shirts of which 48 are good, 8 have major defects and 4 have minor defects. Nigam, a trader, will accept the shirt which are good but Anmol, another trader, will only reject the shirts which have major defects. One shirt is drawn at random from the carton,

(a) Find the probability that it is acceptable to Nigam.

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Conditional Probability

How to handle Dependent Events

Life is full of random events! You need to get a "feel" for them to be a smart and successful person.

Independent Events

Events can be " Independent ", meaning each event is not affected by any other events.

Example: Tossing a coin.

Each toss of a coin is a perfect isolated thing.

What it did in the past will not affect the current toss.

The chance is simply 1-in-2, or 50%, just like ANY toss of the coin.

So each toss is an Independent Event .

Dependent Events

But events can also be "dependent" ... which means they can be affected by previous events ...

Example: Marbles in a Bag

2 blue and 3 red marbles are in a bag.

What are the chances of getting a blue marble?

The chance is 2 in 5

But after taking one out the chances change!

So the next time:

This is because we are removing marbles from the bag.

So the next event depends on what happened in the previous event, and is called dependent .

Replacement

Note: if we replace the marbles in the bag each time, then the chances do not change and the events are independent :

• With Replacement: the events are Independent (the chances don't change)
• Without Replacement: the events are Dependent (the chances change)

Dependent events are what we look at here.

Tree Diagram

A Tree Diagram is a wonderful way to picture what is going on, so let's build one for our marbles example.

There is a 2/5 chance of pulling out a Blue marble, and a 3/5 chance for Red:

We can go one step further and see what happens when we pick a second marble:

If a blue marble was selected first there is now a 1/4 chance of getting a blue marble and a 3/4 chance of getting a red marble.

If a red marble was selected first there is now a 2/4 chance of getting a blue marble and a 2/4 chance of getting a red marble.

Now we can answer questions like "What are the chances of drawing 2 blue marbles?"

Answer: it is a 2/5 chance followed by a 1/4 chance :

Did you see how we multiplied the chances? And got 1/10 as a result.

The chances of drawing 2 blue marbles is 1/10

We love notation in mathematics! It means we can then use the power of algebra to play around with the ideas. So here is the notation for probability:

P(A) means "Probability Of Event A"

In our marbles example Event A is "get a Blue Marble first" with a probability of 2/5:

And Event B is "get a Blue Marble second" ... but for that we have 2 choices:

• If we got a Blue Marble first the chance is now 1/4
• If we got a Red Marble first the chance is now 2/4

So we have to say which one we want , and use the symbol "|" to mean "given":

P(B|A) means "Event B given Event A"

In other words, event A has already happened, now what is the chance of event B?

P(B|A) is also called the "Conditional Probability" of B given A.

And in our case:

P(B|A) = 1/4

So the probability of getting 2 blue marbles is:

And we write it as

"Probability of event A and event B equals the probability of event A times the probability of event B given event A "

Let's do the next example using only notation:

Example: Drawing 2 Kings from a Deck

Event A is drawing a King first, and Event B is drawing a King second.

For the first card the chance of drawing a King is 4 out of 52 (there are 4 Kings in a deck of 52 cards):

P(A) = 4/52

But after removing a King from the deck the probability of the 2nd card drawn is less likely to be a King (only 3 of the 51 cards left are Kings):

P(B|A) = 3/51

P(A and B) = P(A) x P(B|A) =(4/52)x (3/51) = 12/2652 = 1/221

So the chance of getting 2 Kings is 1 in 221, or about 0.5%

Finding Hidden Data

Using Algebra we can also "change the subject" of the formula, like this:

And we have another useful formula:

"The probability of event B given event A equals the probability of event A and event B divided by the probability of event A "

Example: Ice Cream

70% of your friends like Chocolate, and 35% like Chocolate AND like Strawberry.

What percent of those who like Chocolate also like Strawberry?

P(Strawberry|Chocolate) = P(Chocolate and Strawberry) / P(Chocolate)

50% of your friends who like Chocolate also like Strawberry

Big Example: Soccer Game

You are off to soccer, and want to be the Goalkeeper, but that depends who is the Coach today:

• with Coach Sam the probability of being Goalkeeper is 0.5
• with Coach Alex the probability of being Goalkeeper is 0.3

Sam is Coach more often ... about 6 out of every 10 games (a probability of 0.6 ).

So, what is the probability you will be a Goalkeeper today?

Let's build a tree diagram . First we show the two possible coaches: Sam or Alex:

The probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is 1)

Now, if you get Sam, there is 0.5 probability of being Goalie (and 0.5 of not being Goalie):

If you get Alex, there is 0.3 probability of being Goalie (and 0.7 not):

The tree diagram is complete, now let's calculate the overall probabilities. Remember that:

P(A and B) = P(A) x P(B|A)

Here is how to do it for the "Sam, Yes" branch:

(When we take the 0.6 chance of Sam being coach times the 0.5 chance that Sam will let you be Goalkeeper we end up with an 0.3 chance.)

But we are not done yet! We haven't included Alex as Coach:

With 0.4 chance of Alex as Coach, followed by the 0.3 chance gives 0.12

And the two "Yes" branches of the tree together make:

0.3 + 0.12 = 0.42 probability of being a Goalkeeper today

(That is a 42% chance)

One final step: complete the calculations and make sure they add to 1:

0.3 + 0.3 + 0.12 + 0.28 = 1

Yes, they add to 1 , so that looks right.

Friends and Random Numbers

Here is another quite different example of Conditional Probability.

4 friends (Alex, Blake, Chris and Dusty) each choose a random number between 1 and 5. What is the chance that any of them chose the same number?

Let's add our friends one at a time ...

First, what is the chance that Alex and Blake have the same number?

Blake compares his number to Alex's number. There is a 1 in 5 chance of a match.

As a tree diagram :

Note: "Yes" and "No" together  makes 1 (1/5 + 4/5 = 5/5 = 1)

Now, let's include Chris ...

But there are now two cases to consider:

• If Alex and Blake did match, then Chris has only one number to compare to.
• But if Alex and Blake did not match then Chris has two numbers to compare to.

And we get this:

For the top line (Alex and Blake did match) we already have a match (a chance of 1/5).

But for the "Alex and Blake did not match" there is now a 2/5 chance of Chris matching (because Chris gets to match his number against both Alex and Blake).

And we can work out the combined chance by multiplying the chances it took to get there:

Following the "No, Yes" path ... there is a 4/5 chance of No, followed by a 2/5 chance of Yes:

Following the "No, No" path ... there is a 4/5 chance of No, followed by a 3/5 chance of No:

Also notice that when we add all chances together we still get 1 (a good check that we haven't made a mistake):

(5/25) + (8/25) + (12/25) = 25/25 = 1

Now what happens when we include Dusty?

It is the same idea, just more of it:

OK, that is all 4 friends, and the "Yes" chances together make 101/125:

Answer: 101/125

But here is something interesting ... if we follow the "No" path we can skip all the other calculations and make our life easier:

The chances of not matching are:

(4/5) × (3/5) × (2/5) = 24/125

So the chances of matching are:

1 - (24/125) = 101/125

(And we didn't really need a tree diagram for that!)

And that is a popular trick in probability:

It is often easier to work out the "No" case (and subtract from 1 for the "Yes" case)

(This idea is shown in more detail at Shared Birthdays .)

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Probability Assignment

In subject area: Mathematics

A basic probability assignment is a mapping m from the set of all subsets of a universal finite nonempty set X to the interval [0, 1], but it need not meet all the probability measure requirements.

From: North-Holland Series in Applied Mathematics and Mechanics , 2004

Chapters and Articles

You might find these chapters and articles relevant to this topic.

Uncertain Input Data Problems and the Worst Scenario Method

Ivan Hlaváěek Dr. , ... Ivo Babuška Dr. , in North-Holland Series in Applied Mathematics and Mechanics , 2004

Evidence theory

Like a probability measure in probability theory, a basic probability assignment lies in the foundations of evidence theory, also known as the Dempster-Shafer theory ( Dempster, 1967 ), ( Shafer, 1976 ).

A basic probability assignment is a mapping m from the set of all subsets of a universal finite nonempty set X to the interval [0, 1], but it need not meet all the probability measure requirements. Its main features are ∑ S ⊂ X m ( S ) = 1 and m(∅) = 0. The latter equality as well as the finiteness of X are not strictly necessary. Unlike a probability measure, it is not required that S 1 ⊂ S 2 implies m ( S 1 ) ≤ m ( S 2 ) and that m ( S ) + m ( X \ S )= 1.

The set, denoted by F , of focal elements comprises all S ⊂ X such that m ( S ) > 0. The mathematical structure formed by m and F is called a random set on X .

Citing ( Oberkampf et al., 2001 ), we can say that the quantity m ( S )provides a measure of the amount of “likelihood” that is assigned to S . The goal is to exploit such information.

To this end, two mappings from subsets of X to [0, 1] are defined, namely a belief Bel and a plausibility Pl :

Let us again make use of ( Oberkampf et al., 2001 ). The plausibility Pl ( W ) of W provides an upper bound on the likelihood of W , and the belief Bel ( W ) provides a lower bound on the likelihood of W . In one interpretation of belief and plausibility, Bel ( W ) is the smallest probability of W that is consistent with all available evidence, and Pl ( W ) is the largest probability for W that is consistent with all available evidence.

It holds Pl ( W ) = 1 – Bel ( X \ W ), Bel ( W ) = 1 – Pl ( X \ W ).

If Si ⊂ X are focal elements, where i = 1,…, n , and if a map f : X → Z is given, then A j = f ( S i )={ y = f ( x ) : x ∈ S i } and m Z ( A j ) = ∑ { i : f ( S i ) = A j } m ( S i ) introduce focal elements ( j = 1,…, k ≤ n ) and a basic probability assignment into Z . In this way, uncertainty in inputs can be traced and evaluated in model responses. Thus the evidence theory results, in effect, provide bounds on the potential values of the model response that could result from different probability distributions over the intervals that contain input variable values; see ( Oberkampf et al., 2001 ).

We refer to ( Tonon and Bernardini, 1998 ), ( Bernardini, 1999 ), and ( Oberkampf et al., 2001 ) for details, applications, examples, and comparisons with other methods. A study of the combination of evidence, and a large list of various applications of the theory are presented in ( Sentz and Ferson, 2002 ). The mathematical and theoretical approach to the Dempster- Shafer theory from the standpoint of probability theory and decision making under uncertainty is expounded in monograph ( Kramosil, 2001 ).

Judgment Aggregation

Christian List , in Philosophy of Economics , 2012

6.3 Probability aggregation

In the theory of probability aggregation, finally, the focus is not on making consistent acceptance/rejection judgments on the propositions of interest, but rather on arriving at a coherent probability assignment to them (e.g., [ McConway, 1981; Genest and Zidek, 1986; Mongin, 1995 ]). Thus the central question is: How can a group of individuals arrive at a collective probability assignment to a given set of propositions on the basis of the group members' individual probability assignments, while preserving probabilistic coherence (i.e., the satisfaction of the standard axioms of probability theory)? The problem is quite a general one. In a number of decision-making settings, the aim is not so much to come up with acceptance/rejection judgments on certain propositions but rather to arrive at probabilistic information about the degree of belief we are entitled to assign to them or the likelihood of the events they refer to.

Interestingly, the move from a binary to a probabilistic setting opens up some non-degenerate possibilities of aggregation not existent in the standard case of judgment aggregation. A key insight is that probabilistic coherence is preserved under linear averaging of probability assignments. In other words, if each individual coherently assigns probabilities to a given set of propositions, then any weighted linear average of these probability assignments across individuals still constitutes an overall coherent probability assignment. Moreover, it is easy to see that this method of aggregation satisfies the analogues of all the input, output and responsiveness conditions introduced above: i.e., it accepts all possible profiles of coherent individual probability assignments as input, produces a coherent collective probability assignment as output and satisfies the analogues of systematicy and unanimity preservation; it also satisfies anonymity if all individuals are given equal weight in the averaging. A classic theorem by McConway [1981] shows that, if the agenda is isomorphic to a Boolean algebra with more than four elements, linear averaging is uniquely characterized by an independence condition, a unanimity preservation condition as well as the analogues of universal domain and collective rationality. Recently, Dietrich and List [2008b] have obtained a generalization of (a variant of) this theorem for a much larger class of agendas (essentially, the analogue of non-simple agendas). A challenge for the future is to obtain even more general theorems that yield both standard results on judgment aggregation and interesting characterizations of salient probability aggregation methods as special cases.

Causal Inference

D.B. Rubin , in International Encyclopedia of Education (Third Edition) , 2010

The Assignment Mechanism – Formal Notation

A model for the assignment mechanism is needed for all forms of statistical inference for causal effects. Formally, the assignment mechanism gives the conditional probability of each vector of assignments given the covariates and potential outcomes:

A specific example of an assignment mechanism is a completely randomized experiment with N units, where n < N are assigned to the active treatment, and N – n to the control treatment.

An unconfounded assignment mechanism is free of dependence on either Y (0) or Y (1):

With an unconfounded assignment mechanism, at each set of values of X i that has a distinct probability of W i = 1, there is effectively a completely randomized experiment. That is, if X i indicates sex, with males having probability 0.2 of receiving the active treatment and females having probability 0.5 of receiving the active treatment, then essentially one randomized experiment is prescribed for males and another for females.

The assignment mechanism is probabilistic if each unit has a positive probability of receiving either treatment:

where the unit level probabilities are known as propensity scores ( Rosenbaum and Rubin, 1983 ). Unconfounded probabilistic assignment mechanisms often allow particularly straightforward estimation of causal effects from all perspectives, and these assignment mechanisms form the basis for inference for causal effects in more complicated situations, such as when assignment probabilities depend on covariates in unknown ways, or when there is noncompliance with the assigned treatment, or even in observational (nonrandomized) studies. Unconfounded probabilistic assignment mechanisms are essentially generalized randomized experiments, and are called strongly ignorable ( Rosenbaum and Rubin, 1983 ).

A confounded assignment mechanism is one that depends on the potential outcomes. A special class of possibly confounded assignment mechanisms is particularly important to Bayesian inference: ignorable assignment mechanisms ( Rubin, 1978 ). Ignorable assignment mechanisms are defined by their freedom from dependence on any missing potential outcomes:

Ignorable but confounded assignment mechanisms arise in practice, most commonly in sequential experiments, where the next (in time) unit’s probability of being exposed to the active treatment depends on the success rate of those previously exposed to the active treatment versus the success rate of those exposed to the control treatment, as in play-the-winner designs (e.g., Efron, 1971 ). All unconfounded assignment mechanisms are ignorable, but not all ignorable assignment mechanisms are unconfounded (e.g., play-the-winner designs).

Introduction to Probability Theory

Scott L. Miller , Donald Childers , in Probability and Random Processes , 2004

2.3 Assigning Probabilities

In the previous section, probability was defined as a measure of the likelihood of an event or events that satisfy the three Axioms 2.1–2.3. How probabilities are assigned to particular events was not specified. Mathematically, any assignment that satisfies the given axioms is acceptable. Practically speaking, we would like to assign probabilities to events in such a way that the probability assignment actually represents the likelihood of occurrence of that event. Two techniques are typically used for this purpose and are described in the following paragraphs.

In many experiments, it is possible to specify all of the outcomes of the experiment in terms of some fundamental outcomes, which we refer to as atomic outcomes. These are the most basic events that cannot be decomposed into simpler events. From these atomic outcomes, we can build more complicated and more interesting events. Quite often we can justify assigning equal probabilities to all atomic outcomes in an experiment. In that case, if there are M mutually exclusive exhaustive atomic events, then each one should be assigned a probability of 1/ M . Not only does this make perfect common sense, it also satisfies the mathematical requirements of the three axioms that define probability. To see this, we label the M atomic outcomes of an experiment E as ξ 1 , ξ2, …, ξ μ . These atomic events are taken to be mutually exclusive and exhaustive. That is, ξ i ∩ ξ j = θ for all i ≠ j , and ξ 1 ∪ ξ 2 ∪ … ∪ ξ M = S . Then by Corollary 2.1 and Axiom 2.2,

If each atomic outcome is to be equally probable, then we must assign each a probability of Pr(ξ i ) = 1/ M for there to be equality in the preceding equation. Once the probabilities of these outcomes are assigned, the probabilities of some more complicated events can be determined according to the rules set forth in Section 2.2 . This approach to assigning probabilities is referred to as the classical approach.

EXAMPLE 2.6: The simplest example of this procedure is the coin flipping experiment of Example 2.1 . In this case, there are only two atomic events, ξ 1 = H and ξ 2 = T . Provided the coin is fair (again, not biased towards one side or the other), we have every reason to believe that these two events should be equally probable. These outcomes are mutually exclusive and collectively exhaustive (provided we rule out the possibility of the coin landing on its edge). According to our theory of probability, these events should be assigned probabilities of Pr (H) = Pr( T ) = 1/2.

EXAMPLE 2.7: Next consider the dice rolling experiment of Example 2.2 . If the die is not loaded, the six possible faces of the cubicle die are reasonably taken to be equally likely to appear, in which case, the probability assignment is Pr(1) = Pr(2) = ·· ·= Pr(6) = 1/6. From this assignment we can determine the probability of more complicated events, such as

Pr ( even   number   is   rolled ) = Pr ( 2 ∪ 4 ∪ 6 ) = Pr ( 2 ) + Pr ( 4 ) + Pr ( 6 )   ( by   Corollary   2.3 ) = 1 / 6 + 1 / 6 + 1 / 6   ( by   probability   assignment ) = 1 / 2.

EXAMPLE 2.8: In Example 2.3 , a pair of dice were rolled. In this experiment, the most basic outcomes are the 36 different combinations of the six atomic outcomes of the previous example. Again, each of these atomic outcomes is assigned a probability of 1/36. Next, suppose we want to find the probability of the event A = {sum of two dice = 5}. Then,

Pr( A ) = Pr((1,4) ∪ (2,3) ∪ (3,2)∪ (4,1)) = Pr(1,4) + Pr(2,3) + Pr(3,2) + Pr(4,1) (by Corollary 2.1) = 1/36 + 1/36 + 1/36 + 1/36 (by probability assignment) < = 1/9.

Pr ( A ) = Pr ( ( 1 , 4 ) ∪ ( 2 , 3 ) ∪ ( 3 , 2 ) ∪ ( 4 , 1 ) ) = Pr ( 1 , 4 ) + Pr ( 2 , 3 ) + Pr ( 3 , 2 ) + Pr ( 4 , 1 )   ( by   Corollary   2.1 ) = 1 / 36 + 1 / 36 + 1 / 36 + 1 / 36   ( by   probability   assignment ) = 1 / 9.

You should find that each time you run this script, you get different (random) looking results. With any MATLAB command, if you want more information on what the command does, type help followed by the command name at the MATLAB prompt for detailed information on that command. For example, to get help on the rand command, type help rand.

Care must be taken when using the classical approach to assigning probabilities. If we define the set of atomic outcomes incorrectly, unsatisfactory results may occur. In Example 2.8 , we may be tempted to define the set of atomic outcomes as the different sums that can occur on the two die faces. If we assign equally likely probability to each of these outcomes, then we arrive at the assignment

Anyone with experience in games involving dice knows that the likelihood of rolling a 2 is much lower than the likelihood of rolling a 7. The problem here is that the atomic events we have assigned are not the most basic outcomes and can be decomposed into simpler outcomes, as demonstrated in Example 2.8 .

This is not the only problem encountered in the classical approach. Suppose we consider an experiment that consists of measuring the height of an arbitrarily chosen student in your class and rounding that measurement to the nearest inch. The atomic outcomes of this experiment would consist of all the heights of the students in your class. However, it would not be reasonable to assign an equal probability to each height. Those heights corresponding to very tall or very short students would be expected to be less probable than those heights corresponding to a medium height. So, how then do we assign probabilities to these events? The problems associated with the classical approach to assigning probabilities can be overcome by using the relative frequency approach.

The relative frequency approach requires that the experiment we are concerned with be repeatable, in which case, the probability of an event, A , can be assigned by repeating the experiment a large number of times and observing how many times the event, A , actually occurs. If we let n be the number of times the experiment is repeated and n A be the number of times the event A is observed, then the probability of the event A can be assigned according to

This approach to assigning probability is based on experimental results and thus has a more practical flavor to it. It is left as an exercise for the reader (see Exercise 2.6 ) to confirm that this method does indeed satisfy the axioms of probability and is thereby mathematically correct as well.

Table 2.1 . Simulation of Dice Tossing Experiment.

The next to last line of MATLAB code may need some explanation. The double equal sign asks MATLAB to compare the two quantities to see if they are equal. MATLAB responds with 1 for “yes” and 0 for “no.” Hence the expression dice_sum= = 5 results in an n element vector where each element of the vector is either 0 or 1 depending on whether the corresponding element of dice_sum is equal to 5 or not. By summing all elements of this vector, we obtain the number of times the sum 5 occurs in n tosses of the dice.

To get an exact measure of the probability of an event using the relative frequency approach, we must be able to repeat the event an infinite number of times—a serious drawback to this approach. In the dice rolling experiment of Example 2.8 , even after rolling the dice 10,000 times, the probability of observing a 5 was measured to only two significant digits. Furthermore, many random phenomena in which we might be interested are not repeatable. The situation may occur only once, and hence we cannot assign the probability according to the relative frequency approach.

How much to tell your customer? – A survey of three perspectives on selling strategies with incompletely specified products

Jochen Gönsch , in European Journal of Operational Research , 2020

2.5 Assignment timing

Three papers consider the timing of the assignment of the ICSP to its component products and investigate how the seller can benefit from postponing the assignment to the particular customers who bought an ICSP.

Geng's (2016) model is similar to Jiang's (2007) , however he analyzes ICSPs without price discrimination and considers congested systems . The seller provides the two component products at different locations that are each modeled by an M/M/1 queue and can either sell only regular products or only an ICSP. Congestion cost due to waiting is linear in the number of customers and time, and is incurred by the firm. In the baseline model, the firm immediately assigns the component products to customers with equal probability (opaque product). In an extension, the assignment is postponed (flexible product), allowing the firm to operate a single queue and assigning products to customers directly before service provision. Even though price discrimination is not possible, ICSPs can be advantageous.

Although they do not consider assignment timing, we discuss Xu, Lian, Li, and Guo (2016) here, because the paper does not fit in any subsection and is very similar to the aforementioned one. Both were developed apparently independently, but submitted and published at the same time in the same journal. Again, customers have individual tastes according to a Hotelling model for two servers, each with an M/M/1 queue. Waiting costs are incurred by the customers and customers of the probabilistic product are assigned with equal probability to one of the two queues. The authors compare three priority policies, each using its optimal price for regular and probabilistic products: first come first served, high priority for the probabilistic customers, and for the regular ones. They analytically solve the model and show that when the market size is large enough, providing the regular products is sufficient. Otherwise, both the regular and the probabilistic product should be sold. Although the optimal prices depend on the queueing priority policy, the optimal revenue does not.

Wu and Wu (2015) consider demand postponement , motivated by the leisure travel industry in China. In their three period model, a seller provides only one product, but at different points in time. In the advance period, customers can buy the product with guaranteed delivery in the next period, or, at a discount, they can buy an ICSP with delivery in either the next or the following (third) period. Then, the firm makes the inventory decision. In the regular period, guaranteed advance customers are served. When spot customers arrive, they are served subject to availability. Capacity remaining at the end of the regular period is used for buyers of the ICSP. Any remaining buyers of the ICSP are served in the postponement (third) period, where the firm has ample capacity. The authors show that by using postponement, the firm benefits from both stock out cost reduction and capacity waste decrease.

Fay and Xie (2015) explicitly focus on the timing of assignment. Their seller orders inventory upfront to offer two regular products and a probabilistic one. With immediate allocation, the allocation happens after a sale and before the seller learns which product is more popular. Thus, both component products have equal assignment probabilities. With postponed allocation, the decision is made after the sales period when the seller has learnt which product is more popular, and customers (correctly) expect to obtain the less popular one with a higher probability because more inventory from the unpopular product remains. Hence, customers associate a higher expected value with immediate allocation even though they do not care about the delay itself. In this setting, early allocation can be beneficial to the firm, even though it is associated with a higher inventory cost.

Scott L. Miller , Donald Childers , in Probability and Random Processes (Second Edition) , 2012

Section 2.7 Independence

Compare the two probability assignments in Exercises 2.12 and 2.13 . Which of these two assignments corresponds to independent coin tosses?

Find the probability that the first ace is drawn on the 5th selection.

Find the probability that at least 5 cards are drawn before the first ace appears.

Repeat parts (a) and (b) if the cards are drawn without replacement. That is, after each card is drawn, the card is set aside and not replaced in the deck.

Find the probability that the 3rd club is drawn on the 8th selection.

Find the probability that at least 8 cards are drawn before the 3rd club appears.

A computer memory has the capability of storing 10 6 words. Due to outside forces, portions of the memory are often erased. Therefore, words are stored redundantly in various areas of the memory. If a particular word is stored in n different places in the memory, what is the probability that this word cannot be recalled if one-half of the memory is erased by electromagnetic radiation? Hint: Consider each word to be stored in a particular cell (or box).

These cells (boxes) may be located anywhere, geometrically speaking, in memory. The contents of each cell may be either erased or not erased. Assume n is small compared to the memory capacity.

If two events A and B are such that Pr( A ) is not zero and Pr( B ) is not zero, what combinations of independent ( I ), not independent ( NI ), mutually exclusive ( M ), and not mutually exclusive ( NM ) are permissible? In other words, which of the four combinations ( I, M ), ( NI, M ), ( I, NM ), and ( NI, NM ) are permissible? Construct an example for those combinations that are permissible.

Suppose we modify the communications network of Example 2.22 as shown in the diagram by adding a link from node B to node D. Assuming each link is available with probability p independent of any other link, what is the probability of being able to send a message from node A to D?

• Inductive Logic

Maria Carla Galavotti , in Handbook of the History of Logic , 2011

2 The Subjective Interpretation of Probability

Modern subjectivism, sometimes also called “personalism”, shares with logicism the conviction that probability is an epistemic notion. As already pointed out, the crucial point of disagreement between the two interpretations comes in connection with the fact that unlike logicists, subjectivists do not believe that probability evaluations are univocally determined by a given body of evidence.

2.1 The starters

William Fishburn Donkin (1814–1869), professor of astronomy at Oxford, fostered a subjective interpretation of probability in “On Certain Questions Relating to the Theory of Probabilities”, published in 1851. There he writes that “the ‘probability’ which is estimated numerically means merely ‘quantity of belief’, and is nothing inherent in the hypothesis to which it refers” [ Donkin, 1851 , p. 355]. This claim impressed Frank Ramsey, who recorded it in his notes. 28 Donkin's position is actually quite similar to that of De Morgan, especially when he maintains that probability is “ relative to a particular state of knowledge or ignorance; but […] it is absolute in the sense of not being relative to any individual mind; since, the same information being presupposed, all minds ought to distribute their belief in the same way” [ Donkin, 1851 , p. 355]. If in view of claims of this kind Donkin qualifies more as a logicist than as a subjectivist, the appearance of his name in the present section on subjectivism is justified by the fact that he addressed the issue of belief conditioning in a way that anticipated the work of Richard Jeffrey a century later. Donkin formulated a principle, imposing a symmetry restriction on updating belief, as new information is obtained. In a nutshell, the principle states that changing opinion on the probabilities assigned to a set of hypotheses, after new information has been acquired, has to preserve the proportionality among the probabilities assigned to the considered options. Under this condition, the new and old opinions are comparable. The principle is introduced by Donkin as follows:

“ Theorem . If there be any number of mutually exclusive hypotheses, h 1 , h 2 , h 3 …, of which the probabilities relative to a particular state of information are p 1 , p 2 , p 3 …, and if new information be gained which changes the probabilities of some of them, suppose of h m +1 and all that follow, without having otherwise any reference to the rest , then the probabilities of these latter have the same ratios to one another, after the new information, that they had before ; that is, p ′ 1 : p ′ 2 : p ′ 3 : … : p ′ m = p 1 : p 2 : p 3 : … : p m , where the accented letters denote the values after the new information has been acquired”. [ Donkin, 1851 , p. 356]

The method of conditioning known as Jeffrey conditionalization reflects precisely the intuition behind Donkin's principle. 29

The French mathematician Émile Borel (1871–1956), who gave outstanding contributions to the study of the mathematical properties of probability, can be considered a pioneer of the subjective interpretation. In a review of Keynes' Treatise originally published in 1924 and later reprinted in the last volume of the series of monographs edited by Borel under the title Traité du calcul des probabilités et ses applications (1939), 30 Borel raises various objections to Keynes, blamed for overlooking the applications of probability to science to focus only on the probability of judgments. Borel takes this to be a distinctive feature of the English as opposed to continental literature which he regards as more aware of the developments of science, particularly physics. When making such claims, Borel is likely to have in mind above all Henri Poincaré, whose ideas exercised a certain influence on him. 31

While agreeing with Keynes in taking probability in its epistemic sense, Borel claims that probability acquires a different meaning depending on the context in which it occurs. Probability has a different value in situations characterized by a different state of information, and is endowed with a “more objective” meaning in science, where its assessment is grounded on a strong body of information, shared by the scientific community.

Borel is definitely a subjectivist when he admits that two people, given the same information, can come up with different probability evaluations. This is most common in everyday applications of probability, like horse races, or weather forecasts. In all such cases, probability judgments are of necessity relative to “a certain body of knowledge”, which is not the kind of information shared by everyone, like scientific theories at a certain time. Remarkably, Borel maintains that when talking of this kind of probability the “body of knowledge” in question should be thought of as “necessarily included in a determinate human mind, but not such that the same abstract knowledge constitutes the same body of knowledge in two distinct human minds” [ Borel, 1924 , English edition 1964, p. 51]. Probability evaluations made at different times, based on different information, ought not be taken as refinements of previous judgments, but as totally new ones.

Borel disagrees with Keynes on the claim that there are probabilities which cannot be evaluated numerically. In connection with the evaluation of probability Borel appeals to the method of betting, which “permits us in the majority of cases a numerical evaluation of probabilities” [ Borel, 1924 , English edition 1964, p. 57]. This method, which dates back to the origin of the numerical notion of probability in the seventeenth century, is regarded by Borel as having

“exactly the same characteristics as the evaluation of prices by the method of exchange. If one desires to know the price of a ton of coal, it suffices to offer successively greater and greater sums to the person who possesses the coal; at a certain sum he will decide to sell it. Inversely if the possessor of the coal offers his coal, he will find it sold if he lowers his demands sufficiently”. [ Borel, 1924 , English edition 1964, p. 57]

At the end of a discussion of the method of bets, where he takes into account some of the traditional objections against it, Borel concludes that this method seems good enough, in the light of ordinary experience.

Borel's conception of epistemic probability has a strong affinity with the subjective interpretation developed by Ramsey and de Finetti. In a brief note on Borel's work, de Finetti praises Borel for holding that probability must be referred to the single case, and that this kind of probability is always measurable sufficiently well by means of the betting method. At the same time, de Finetti strongly disagrees with the eclectic attitude taken by Borel, more particularly with his admission of an objective meaning of probability, in addition to the subjective. 32

2.2 Ramsey and the principle of coherence

Frank Plumpton Ramsey (1903–1930), Fellow of King's College and lecturer in mathematics at Cambridge, made outstanding contributions to a number of different fields, including mathematics, logic, philosophy, probability, and economics. 33 In his obituary, Keynes refers to Ramsey's as “one of the brightest minds of our generation” and praises him for the “amazing, easy efficiency of the intellectual machine which ground away behind his wide temples and broad, smiling face” [ Keynes, 1930, 1972 , p. 336]. A regular attender at the meetings of the Moral Sciences Club and the Apostles, Ramsey actively interacted with his contemporaries, including Keynes, Moore, Russell and Wittgenstein — whose Tractatus he translated into English — often influencing their ideas.

Ramsey is considered the starter of modern subjectivism with his paper “Truth and Probability”, read at the Moral Sciences Club in 1926, and published in 1931 in the collection The Foundations of Mathematics and Other Logical Essays edited by Richard Bevan Braithwaite shortly after Ramsey's death. Other sources are to be found in the same book, as well as in the other collection, edited by Hugh Mellor, Philosophical Papers (largely overlapping Braithwaite's), and in addition in the volumes Notes on Philosophy, Probability and Mathematics , edited by Maria Carla Galavotti, and On Truth , edited by Nicholas Rescher and Ulrich Majer.

Ramsey regards probability as a degree of belief, and probability theory as a logic of partial belief. Degree of belief is taken as a primitive notion having “no precise meaning unless we specify more exactly how it is to be measured” [ Ramsey, 1990a , p. 63]; in other words, degree of belief requires an operative definition that specifies how it can be measured. A “classical” way of measuring degree of belief is the method of bets, endowed with a long-standing tradition dating back to the birth of probability in the seventeenth century with the work of Blaise Pascal, Pierre Fermat and Christiaan Huygens. In Ramsey's words: “the old established way of measuring a person's belief is to propose a bet, and see what are the lowest odds which he will accept” ( Ramsey [1990a ], p. 68). Such a method, however, suffers from well known problems, like the diminishing marginal utility of money, and is to a certain extent arbitrary, due to personal “eagerness or reluctance to bet”, and the fact that “the proposal of a bet may inevitably alter” a person's “state of opinion” ( Ramsey [1990a] , p. 68).

To avoid such difficulties, Ramsey adopted an alternative method based on the notion of preference , grounded in a “general psychological theory” asserting that “we act in the way we think most likely to realize the objects of our desires, so that a person's actions are completely determined by his desires and opinions” [ Ramsey, 1990a , p. 69]. Attention is called to the fact that

“this theory is not to be identified with the psychology of the Utilitarians, in which pleasure had a dominant position. The theory I propose to adopt is that we seek things which we want, which may be our own or other people's pleasure, or anything else whatever, and our actions are such as we think most likely to realize these goods.” [ Ramsey, 1990a , p. 69]

After clarifying that “good” and “bad” are not to be taken in an ethical sense, “but simply as denoting that to which a given person feels desire and aversion” [ Ramsey, 1990a , p. 70], Ramsey introduces the notion of quantity of belief , by assuming that goods are measurable as well as additive, and that an agent “will always choose the course of action which will lead in his opinion to the greatest sum of good” [Ramsey, [990a, p. 70]. The fact that people hardly ever entertain a belief with certainty, and usually act under uncertainty, is accounted for by appealing to the principle of mathematical expectation, which Ramsey introduces “as a law of psychology”. Given a person who is prepared to act in order to achieve some good,

“if p is a proposition about which he is doubtful, any goods or bads for whose realization p is in his view a necessary and sufficient condition enter into his calculation multiplied by the same fraction, which is called the ‘degree of his belief in p ’. We thus define degree of belief in a way which presupposes the use of mathematical expectation”. [ Ramsey, 1990a , p. 70]

An alternative definition of degree of belief is also suggested along the following lines: “Suppose [the] degree of belief [of a certain person] in p is m / n ; then his action is such as he would choose it to be if he had to repeat it exactly n times, in m of which p was true, and in the others false” [ Ramsey, 1990a , p. 70]. The two accounts point out two different, albeit strictly intertwined, aspects of the same concept, and are taken to be equivalent.

Ramsey exemplifies a typical situation involving a choice of action that depends on belief as follows:

“I am at a cross-roads and do not know the way; but I rather think one of the two ways is right. I propose therefore to go that way but keep my eyes open for someone to ask; if now I see someone half a mile away over the fields, whether I turn aside to ask him will depend on the relative inconvenience of going out of my way to cross the fields or of continuing on the wrong road if it is the wrong road. But it will also depend on how confident I am that I am right; and clearly the more confident I am of this the less distance I should be willing to go from the road to check my opinion. I propose therefore to use the distance I would be prepared to go to ask, as a measure of the confidence of my opinion”. [ Ramsey, 1990a , pp. 70–71]

Denoting f ( x ) the disadvantage of walking x metres, r the advantage of reaching the right destination, and w the disadvantage of arriving at a wrong destination, if I were ready to go a distance d to ask, the degree of belief that I am on the right road is p = 1 − ( f ( d )/( r − w )). To choose an action of this kind can be considered advantageous if, were I to act n times in the same way, np times out of these n I was on the right road (otherwise I was on the wrong one). In fact, the total good of not asking each time is npr + n (1 − p ) w = nw + np ( r − w ); while the total good of asking each time (in which case I would never go wrong) is nr − nf ( x ). The total good of asking is greater than the total good of not asking, provided that f ( x ) ≺ ( r − w )(1 − p ). Ramsey concludes that the distance d is connected with my degree of belief, p , by the relation f ( d ) = ( r − w )(1 − p ), which amounts to p = 1 − ( f ( d )/( r − w )), as stated above. He then observes that

“It is easy to see that this way of measuring beliefs gives results agreeing with ordinary ideas. […] Further, it allows validity to betting as means of measuring beliefs. By proposing to bet on p we give the subject a possible course of action from which so much extra good will result to him if p is true and so much extra bad if p is false”. [ Ramsey, 1990a , p. 72]

However, given the already mentioned difficulties connected with the betting scheme, Ramsey turns to a more general notion of preference . Degree of belief is then operationally defined in terms of personal preferences, determined on the basis of the expectation of an individual of obtaining certain goods, not necessarily of a monetary kind. The value of such goods is intrinsically relative, because they are defined with reference to a set of alternatives. The definition of degree of belief is committed to a set of axioms, which provide a way of representing its values by means of real values. Degrees of belief obeying such axioms are called consistent . The laws of probability are then spelled out in terms of degrees of belief, and it is argued that consistent sets of degrees of belief satisfy the laws of probability. Additivity is assumed in a finite sense, since the set of alternatives taken into account is finite. In this connection Ramsey observes that the human mind is only capable of contemplating a finite number of alternatives open to action, and even when a question is conceived, allowing for an infinite number of answers, these have to be lumped “into a finite number of groups” [ Ramsey, 1990a , p. 79].

The crucial feature of Ramsey's theory of probability is the link between probability and degree of belief established by consistency , or coherence — to use the term that is commonly adopted today. Consistency guarantees the applicability of the notion of degree of belief, which can therefore qualify as an admissible interpretation of probability. In Ramsey's words, the laws of probability can be shown to be

“necessarily true of any consistent set of degrees of belief. Any definite set of degrees of belief which broke them would be inconsistent in the sense that it violated the laws of preference between options. […] If anyone's mental condition violated these laws, his choice would depend on the precise form in which the options were offered him, which would be absurd. He could have a book made against him by a cunning better and would then stand to lose in any event. We find, therefore, that a precise account of the nature of partial belief reveals that the laws of probability are laws of consistency. […] Having any definite degree of belief implies a certain measure of consistency, namely willingness to bet on a given proposition at the same odds for any stake, the stakes being measured in terms of ultimate values. Having degrees of belief obeying the laws of probability implies a further measure of consistency, namely such a consistency between the odds acceptable on different propositions as shall prevent a book being made against you”. [ Ramsey, 1990a , p. 78]

By arguing that from the assumption of coherence one can derive the laws of probability Ramsey paved the way to a fully-fledged subjectivism. Remarkably, within this perspective the laws of probability “do not depend for their meaning on any degree of belief in a proposition being uniquely determined as the rational one; they merely distinguish those sets of beliefs which obey them as consistent ones” [ Ramsey, 1990a , p. 78]. This claim brings us to the core of subjectivism, for which coherence is the only condition that degrees of belief should obey, or, to put it slightly differently, insofar as a set of degrees of belief is coherent there is no further demand of rationality to be met.

Having adopted a notion of probability in terms of coherent degrees of belief, Ramsey does not need to rely on the principle of indifference. In his words: “the Principle of Indifference can now be altogether dispensed with” [ Ramsey, 1990a , p. 85]. This is a decisive step in the moulding of modern subjectivism. As we will see in the next Section, a further step was made by Bruno de Finetti, who supplied the “static” definition of subjective probability in terms of coherent degrees of belief with a “dynamic” dimension, obtained by joining subjective probability with exchangeability within the framework of the Bayesian method. 34 Although this crucial step was actually made by de Finetti, there is evidence that Ramsey knew the property of exchangeability, of which he must have heard from Johnson's lectures. Evidence for this claim is found in his note “Rule of Succession”, where use is made of the notion of exchangeability, named “equiprobability of all permutations”. 35 What apparently Ramsey did not see, and was instead grasped by de Finetti, is the usefulness of applying exchangeability to the inductive procedure, modelled upon Bayes' rule. Remarkably, in another note called “Weight or the Value of Knowledge”, 36 Ramsey was able to prove that collecting evidence pays in expectation, provided that acquiring the new information is free, and shows how much the increase in weight is. This shows he had a dynamic view at least of this important process. As pointed out by Nils-Eric Sahlin and Brian Skyrms, Ramsey's note on weight anticipates subsequent work by Savage, Good, and others. 37

Ramsey put forward his theory of probability in open contrast with Keynes. In particular, Ramsey did not share Keynes' claim that “a probability may […] be unknown to us through lack of skill in arguing from given evidence” [ Ramsey, 1922 ,1989, p. 220]. For a subjectivist, the notion of unknown probability does not make much sense, as repeatedly emphasized also by de Finetti. Moreover, Ramsey criticized the logical relations on which Keynes' theory rests. In “Criticism of Keynes” he writes that: “There are no such things as these relations. a) Do we really perceive them? Least of all in the simplest cases when they should be clearest; can we really know them so little and yet be so certain of the laws which they testify? […] c) They would stand in such a strange correspondence with degrees of belief” [ Ramsey, 1991a , pp. 273–274].

Like Keynes, Ramsey believed that probability is the object of logic, but they disagreed on the nature of that logic. Ramsey distinguished between a “lesser logic, which is the logic of consistency, or formal logic”, and a “larger logic, which is the logic of discovery, or inductive logic” [ Ramsey, 1990a , p. 82]. The “lesser” logic, which is the logic of tautologies in Wittgenstein's sense, can be “interpreted as an objective science consisting of objectively necessary propositions”. By contrast, the “larger” logic, which includes probability, does not share this feature, because “when we extend formal logic to include partial beliefs this direct objective interpretation is lost” [ Ramsey, 1990a , p. 83], and can only be endowed with a psychological foundation . 38 Ramsey's move towards psychologism was inspired by Wittgenstein. This is manifest in a paper read to the Apostles in 1922, called “Induction: Keynes and Wittgenstein”, where Wittgenstein's psychologism is contrasted with Keynes' logicism. At the beginning of that paper, Ramsey mentions propositions 6.363 and 6.3631 of the Tractatus , where it is maintained that the process of induction “has no logical foundation but only a psychological one” [ Ramsey, 1991a , p. 296]. After praising Wittgenstein for his appeal to psychology in order to justify the inductive procedure, Ramsey discusses Keynes' approach at length, expressing serious doubts on his attempt at grounding induction on logical relations and hypotheses. At the end of the paper, after recalling Hume's celebrated argument, Ramsey puts forward by way of a conjecture, of which he claims to be too tired “to see clearly if it is sensible or absurd”, the idea that induction could be justified by saying that

“a type of inference is reasonable or unreasonable according to the relative frequencies with which it leads to truth and falsehood. Induction is reasonable because it produces predictions which are generally verified, not because of any logical relation between its premisses and conclusions. On this view we should establish by induction that induction was reasonable, and induction being reasonable this would be a reasonable argument”. [ Ramsey, 1991a , p. 301]

This passage suggests that Ramsey had in mind a pragmatic justification of the inductive procedure. A similar attitude reappears at the end of “Truth and Probability”, where he describes his own position as “a kind of pragmatism”, holding that

“we judge mental habits by whether they work, i.e. whether the opinions they lead to are for the most part true, or more often true than those which alternative habits would lead to. Induction is such a useful habit, and so to adopt it is reasonable. All that philosophy can do is to analyse it, determine the degree of its utility, and find on what characteristics of nature it depends. An indispensable means for investigating these problems is induction itself, without which we should be helpless. In this circle lies nothing vicious. It is only through memory that we can determine the degree of accuracy of memory; for if we make experiments to determine this effect, they will be useless unless we remember them”. [ Ramsey, 1990a , p. 93–94]

As testified by a number of Ramsey's references to William James and Charles Sanders Peirce, pragmatism is a major feature of his philosophy in general, and his views on probability are no exception.

A puzzling aspect of Ramsey's theory of probability are the relations between degree of belief and frequency. In “Truth and Probability” he writes that “it is natural […] that we should expect some intimate connection between these two interpretations, some explanation of the possibility of applying the same mathematical calculus to two such different sets of phenomena” [ Ramsey, 1990a , p. 83]. Such a connection is identified with the fact that “the very idea of partial belief involves reference to a hypothetical or ideal frequency […] belief of degree m / n is the sort of belief which leads to the action which would be best if repeated n times in m of which the proposition is true” [ Ramsey, 1990a , p. 84].

This passage — echoing the previously mentioned conjecture from “Induction: Keynes and Wittgenstein” — reaffirms Ramsey's pragmatical tendency to associate belief with action, and to justify inductive behaviour with reference to successful conduct. The argument is pushed even further when Ramsey says that

“It is this connection between partial belief and frequency which enables us to use the calculus of frequencies as a calculus of consistent partial belief. And in a sense we may say that the two interpretations are the objective and subjective aspects of the same inner meaning, just as formal logic can be interpreted objectively as a body of tautology and subjectively as the laws of consistent thought”. [ Ramsey, 1990a , p. 84]

However, in other passages the connection between these two “aspects” is not quite so strict:

“experienced frequencies often lead to corresponding partial beliefs, and partial beliefs lead to the expectation of corresponding frequencies in accordance with Bernoulli's Theorem. But neither of these is exactly the connection we want; a partial belief cannot in general be connected uniquely with any actual frequency”. [ Ramsey, 1990a , p. 84]

Evidence that Ramsey was intrigued by the relation between frequency and degree of belief is found in some remarks contained in the note “Miscellaneous Notes on Probability”, written in 1928. There four kinds of connections are pointed out, namely: “(1) if degree of belief = γ, most prob((able)) frequency is γ (if instances independent). This is Bernoulli's theorem; (2) if freq((uency)) has been γ we tend to believe with degree γ; (3) if freq((uency)) is γ, degree γ of belief is justified. This is Peirce's definition; (4) degree γ of belief means acting appropriately to a frequency γ” [ Ramsey, 1991a , p. 275]. After calling attention to such possible connections, Ramsey reaches the conclusion that “it is this last which makes calculus of frequencies applicable to degrees of belief”. Remarkably, the result known as de Finetti's “representation theorem” tells us precisely how to treat relation (4). One might speculate that Ramsey would have found an answer to at least part of what he was looking for in this result, that de Finetti found out in the very same years, but was not available to him. 39

Claims like that mentioned above to the effect that partial belief and frequency “are the two objective and subjective aspects of the same inner meaning”, might be taken to suggest that Ramsey admitted of two notions of probability: one epistemic (the subjective view) and one empirical (the frequency view). 40 This emerges again at the very beginning of “Truth and Probability” where Ramsey claims that although the paper deals with the logic of partial belief, “there is no intention of implying that this is the only or even the most important aspect of the subject”, adding that “probability is of fundamental importance not only in logic but also in statistical and physical science, and we cannot be sure beforehand that the most useful interpretation of it in logic will be appropriate in physics also” [ Ramsey, 1990a , p. 53]. It can be argued that in spite of these claims Ramsey trusted that the subjective interpretation has the resources for accounting for all uses of probability. His writings offer plenty of evidence for this thesis.

There is no doubt that Ramsey took seriously the problem of what kind of probability is employed in science. We know from Braithwaite's “Introduction” to The Foundations of Mathematics that he had planned to write a final section of “Truth and Probability”, dealing with probability in science. We also know from Ramsey's unpublished notes that by the time of his death he was working on a book bearing the title “On Truth and Probability”, of which he left a number of tables of contents. 41 Of the projected book he only wrote the first part, dealing with the notion of truth, which was published in 1991 under the title On Truth . It can be conjectured that he meant to include in the second part of the book the content of the paper “Truth and Probability”, plus some additional material on probability in science. The notes published in The Foundations of Mathematics under the heading “Further Considerations”, 42 and a few more published in the volume Notes on Philosophy, Probability and Mathematics , contain evidence that in the years 1928–29 Ramsey was actively thinking about such problems as theories, laws, causality, chance, all of which he regarded as intertwined. A careful analysis of such writings shows that — contrary to the widespread opinion that he was a dualist with regard to probability — in the last years of his life Ramsey was developing a view of chance and probability in physics fully compatible with his subjective interpretation of probability as degree of belief.

Ramsey's view of chance revolves around the idea that this notion requires some reference to scientific theories. Chance cannot be defined simply in terms of laws (empirical regularities) or frequencies — though the specification of chances involves reference to laws, in a way that will soon be clarified. In “Reasonable Degree of Belief” Ramsey writes that “We sometimes really assume a theory of the world with laws and chances and mean not the proportion of actual cases but what is chance on our theory” [ Ramsey, 1990a , p. 97]. The same point is emphasized in the note “Chance”, also written in 1928, where the frequency-based views of chance put forward by authors like Norman Campbell is criticized. The point is interesting, because it highlights Ramsey's attitude to frequentism, which, far from considering a viable interpretation of probability, he deems inadequate. As Ramsey puts it:

“There is, for instance, no empirically established fact of the form ‘In n consecutive throws the number of heads lies between n /2±ε( n )’. On the contrary we have good reason to believe that any such law would be broken if we took enough instances of it. Nor is there any fact established empirically about infinite series of throws; this formulation is only adopted to avoid contradiction by experience; and what no experience can contradict, none can confirm, let alone establish”. [ Ramsey, 1990a , p. 104]

To Campbell's frequentist view, Ramsey opposed a notion of chance ultimately based on degrees of belief. He defines it as follows:

“Chances are degrees of belief within a certain system of beliefs and degrees of belief; not those of any actual person, but in a simplified system to which those of actual people, especially the speaker, in part approximate. […] This system of beliefs consists, firstly, of natural laws, which are in it believed for certain, although, of course, people are not really quite certain of them”. [ Ramsey, 1990a , p. 104]

In addition, the system will contain statements of the form: “when knowing ψx and nothing else relevant, always expect ϕx with degree of belief p (what is or is not relevant is also specified in the system)” [ Ramsey, 1990a , p. 104]. Such statements together with the laws “form a deductive system according to the rules of probability, and the actual beliefs of a user of the system should approximate to those deduced from a combination of the system and the particular knowledge of fact possessed by the user, this last being (inexactly) taken as certain” [ Ramsey, 1990a , p. 105]. To put it differently, chance is defined with reference to systems of beliefs that typically contain accepted laws.

Ramsey stresses that chances “must not be confounded with frequencies”, for the frequencies actually observed do not necessarily coincide with them. Unlike frequencies, chances can be said to be “objective” in two ways. First, to say that a system includes a chance value referred to a phenomenon, means that the system itself cannot be modified so as to include a pair of deterministic laws, ruling the occurrence and non-occurrence of the same phenomenon. As explicitly admitted by Ramsey, this characterization of objective chance is reminiscent of Poincaré's treatment of the matter, and typically applies “when small causes produce large effects” [ Ramsey, 1990a , p. 106]. Second, chances can be said to be objective “in that everyone agrees about them, as opposed e.g. to odds on horses” [ Ramsey, 1990a , p. 106)].

On the basis of this general definition of chance, Ramsey qualifies probability in physics as chance referred to a more complex system, namely to a system making reference to scientific theories. In other words, probabilities occurring in physics are derived from physical theories. They can be taken as ultimate chances , to mean that within the theoretical framework in which they occur there is no way of replacing them with deterministic laws. The objective character of chances descends from the objectivity peculiarly ascribed to theories that are universally accepted.

Ramsey's view of chance and probability in physics is obviously intertwined with his conception of theories, truth and knowledge in general. Within Ramsey's philosophy the “truth” of theories is accounted for in pragmatical terms. In this connection Ramsey holds the view, whose paternity is usually attributed to Charles Sanders Peirce, but is also found in Campbell's work, that theories which gain “universal assent” in the long run are accepted by the scientific community and taken as true. Along similar lines he characterized a “true scientific system” with reference to a system to which the opinion of everyone, grounded on experimental evidence, will eventually converge. According to this pragmatically oriented view, chance attributions, like all general propositions belonging to theories — including causal laws — are not to be taken as propositions, but rather as “variable hypotheticals”, or “rules for judging”, apt to provide a tool with which the user meets the future. 43

To sum up, for Ramsey chances are theoretical constructs, but they do not express realistic properties of “physical objects”, whatever meaning be attached to this expression. Chance attributions indicate a way in which beliefs in various facts belonging to science are guided by scientific theories. Ramsey's idea that within the framework of subjective probability one can make sense of an “objective” notion of physical probability has passed almost unnoticed. It is, instead, an important contribution to the subjective interpretation and its possible applications to science.

2.3 de Finetti and exchangeability

With the Italian Bruno de Finetti (1906–1985) the subjective interpretation of probability came to completion. Working in the same years as Ramsey, but independently, de Finetti forged a similar view of probability as degree of belief, subject to the only constraint of coherence. To such a definition he added the notion of exchangeability, which can be regarded as the decisive step towards the edification of modern subjectivism. In fact exchangeability, combined with Bayes' rule, gives rise to the inferential methodology which is at the root of the so-called neo-Bayesianism. This result was the object of the paper “Funzione caratteristica di un fenomeno aleatorio” that de Finetti read at the International Congress of Mathematicians, held in Bologna in 1928. In 1935, at Maurice Fréchet's invitation de Finetti gave a series of lectures at the Institut Henri Poincaré in Paris, whose text was published in 1937 under the title “La prévision: ses lois logiques, ses sources subjectives”. This article, which is one of de Finetti's best known, allowed dissemination of his ideas in the French speaking community of probabilists. However, de Finetti's work came to be known to the English speaking community only in the 1950s, thanks to Leonard Jimmie Savage, with whom he entertained a fruitful collaboration. In addition to making a contribution to probability theory and statistics which is universally recognized as seminal, de Finetti put forward an original philosophy of probability, which can be described as a blend of pragmatism, operationalism and what we would today call “anti-realism”. 44 indexAliotta, A.

Richard Jeffrey labelled de Finetti's philosophical position “radical probabilism” 45 to stress the fact that for de Finetti probability imbues the whole edifice of human knowledge, and that scientific knowledge is a product of human activity ruled by (subjective) probability, rather than truth or objectivity. De Finetti's outlined his philosophy of probability in the article “Probabilismo” (1931) which he regarded as his philosophical manifesto. Yet another philosophical text bearing the title L'invenzione della verità , originally written by de Finetti in 1934 to take part in a competition for a grant from the Royal Academy of Italy, was published in 2006. The two main sources of de Finetti's philosophy are Mach's phenomenalism, and pragmatism, namely the version upheld by the so-called Italian pragmatists, including Giovanni Vailati, Antonio Aliotta and Mario Calderoni. The starting point of de Finetti's probabilism is the refusal of the notion of truth, and the related view that there are “immutable and necessary” laws. In “Probabilismo” he writes:

“no science will permit us to say: this fact will come about, it will be thus and so because it follows from a certain law, and that law is an absolute truth. Still less will it lead us to conclude skeptically: the absolute truth does not exist, and so this fact might or might not come about, it may go like this or in a totally different way, I know nothing about it. What we can say is this: I foresee that such a fact will come about, and that it will happen in such and such a way, because past experience and its scientific elaboration by human thought make this forecast seem reasonable to me”. [ de Finetti, 1931a , English edition 1989, p. 170]

Probability makes forecast possible, and since a forecast is always referred to a subject, being the product of his experience and convictions, the instrument we need is the subjective theory of probability. For de Finetti probabilism is the way out of the antithesis between absolutism and skepticism, and at its core lies the subjective notion of probability. Probability “means degree of belief (as actually held by someone, on the ground of his whole knowledge, experience, information) regarding the truth of a sentence , or event E (a fully specified ‘single’ event or sentence, whose truth or falsity is, for whatever reason, unknown to the person)” [ de Finetti, 1968 , p. 45]. Of this notion, de Finetti wants to show not only that it is the only non contradictory one, but also that it covers all uses of probability in science and everyday life. This programme is accomplished in two steps: first, an operational definition of probability is worked out, second, it is argued that the notion of objective probability is reducible to that of subjective probability.

As we have seen discussing Ramsey's theory of probability, the obvious option to define probability in an operational fashion is in terms of betting quotients. Accordingly, the degree of probability assigned by an individual to a certain event is identified with the betting quotient at which he would be ready to bet a certain sum on its occurrence. The individual in question should be thought of as one in a condition to bet whatever sum against any gambler whatsoever, free to choose the betting conditions, like someone holding the bank at a gambling-casino. Probability is defined as the fair betting quotient he would attach to his bets. De Finetti adopts this method, with the proviso that in case of monetary gain only small sums should be considered, to avoid the problem of marginal utility. Like Ramsey, de Finetti states coherence as the fundamental and unique criterion to be obeyed to avoid a sure loss, and spells out an argument to the effect that coherence is a sufficient condition for the fairness of a betting system, showing that a coherent gambling behaviour satisfies the principles of probability calculus, which can be derived from the notion of coherence itself. This is known in the literature as the Dutch book argument.

It is worth noting that for de Finetti the scheme of bets is just a convenient way of making probability readily understandable, but he always held that there are other ways of defining probability. In “Sul significato soggettivo della probabilità” [ de Finetti, 1931b ], after giving an operational definition of probability in terms of coherent betting systems, de Finetti introduces a qualitative definition of subjective probability based on the relation of “at least as probable as”. He then argues that it is not essential to embrace a quantitative notion of probability, and that, while betting quotients are apt devices for measuring and defining probability in an operational fashion, they are by no means an essential component of the notion of probability, which is in itself a primitive notion, expressing “an individual's psychological perception” [ de Finetti, 1931b , English edition 1992, p. 302]. The same point is stressed in Teoria delle probabilità , where de Finetti describes the betting scheme as a handy tool leading to “simple and useful insights” [ de Finetti, 1970 , English edition 1975, vol. 1, p. 180], but introduces another method of measuring probability, making use of scoring rules based on penalties. Remarkably, de Finetti assigns probability an autonomous value independent from the notion utility, thereby marking a difference between his position and that of Ramsey and other supporters of subjectivism, like Savage.

The second step of de Finetti's programme, namely the reduction of objective to subjective probability, relies on what is known as the “representation theorem”. The pivotal notion in this context is that of exchangeability , which corresponds to Johnson's “permutation postulate” and Carnap's “symmetry”. 46 Summarizing de Finetti, events belonging to a sequence are exchangeable if the probability of h successes in n events is the same, for whatever permutation of the n events, and for every n and h ≤ n . The representation theorem says that the probability of exchangeable events can be represented as follows. Imagine the events were probabilistically independent, with a common probability of occurrence p . Then the probability of a sequence e , with h occurrences in n , would be p h (1 − p ) n−h . But if the events are exchangeable, the sequence has a probability P ( e ), represented according to de Finetti's representation theorem as a mixture over the p h (1 − p ) n−h with varying values of p :

where the distribution function F ( p ) is unique. The above equation involves two kinds of probability, namely the subjective probability P ( e ) and the “objective” (or “unknown”) probability p of the events considered. This enters into the mixture associated with the weights assigned by the function F ( p ) representing a probability distribution over the possible values of p . Assuming exchangeability then amounts to assuming that the events considered are equally distributed and independent, given any value of p .

In order to understand de Finetti's position, it is useful to start by considering how an objectivist would proceed when assessing the probability of an unknown event. An objectivist would assume an objective success probability p . But its value would in general remain unknown. One could give weights to the possible values of p , and determine the weighted average. The same applies to the probability of a sequence e , with h successes in n independent repetitions. Note that because of independence it does not matter where the successes appear. De Finetti focuses on the latter, calling exchangeable those sequences where the places of successes do not make a difference in probability. These need not be independent sequences. An objectivist who wanted to explain subjective probability, would say that the weighted averages are precisely the subjective probabilities. But de Finetti proceeds in the opposite direction with his representation theorem: starting from the subjective judgment of exchangeability, one can show that there is only one way of giving weights to the possible values of the unknown objective probabilities. According to this interpretation, objective probabilities become useless and subjective probability can do the whole job. De Finetti holds that exchangeability represents the correct way of expressing the idea that is usually conveyed by the expression “independent events with constant but unknown probability”. If we take an urn of unknown composition, says de Finetti, the above phrase means that, relative to each of all possible compositions of the urn, the events can be seen as independent with constant probability. Then he points out that

“what is unknown here is the composition of the urn, not the probability: this latter is always known and depends on the subjective opinion on the composition, an opinion which changes as new draws are made and the observed frequency is taken into account”. [ de Finetti, 1995 , English edition 2008, p. 163]

It should not pass unnoticed that for the subjectivist de Finetti probability, being the expression of the feelings of the subjects who evaluate it, is always definite and known.

From a philosophical point of view, de Finetti's reduction of objective to subjective probability is to be seen pragmatically; it follows the same pragmatic spirit inspiring the operational definition of subjective probability, and complements it. From a more general viewpoint, the representation theorem gives applicability to subjective probability, by bridging the gap between degrees of belief and observed frequencies. Taken in connection with Bayes' rule, exchangeability provides a model of how to proceed in such a way as to allow for an interplay between the information on frequencies and degrees of belief. By showing that the adoption of Bayes' method, taken in conjunction with exchangeability, leads to a convergence between degrees of belief and frequencies, de Finetti indicates how subjective probability can be applied to statistical inference.

According to de Finetti, the representation theorem answers Hume's problem because it justifies “why we are also intuitively inclined to expect that frequency observed in the future will be close to frequency observed in the past” [ de Finetti, 1972a , p. 34]. De Finetti's argument is pragmatic and revolves around the task of induction: to guide inductive reasoning and behavior in a coherent way. Like Hume, de Finetti thinks that it is impossible to give a logical justification of induction, and answers the problem in a psychologistic fashion.

De Finetti's probabilism is deeply Bayesian: to his eyes statistical inference can be entirely performed by exchangeability in combination with Bayes' rule. From this perspective, the shift from prior to posterior, or, as he preferred to say, from initial to final probabilities, becomes the cornerstone of statistical inference. In a paper entitled “Initial Probabilities: a Prerequisite for any Valid Induction” de Finetti takes a “radical approach” by which “all the assumptions of an inference ought to be interpreted as an overall assignment of initial probabilities” [ de Finetti, 1969 , p. 9]. The shift from initial to final probabilities receives a subjective interpretation, in the sense that it means going from one subjective probability to another, although objective factors, like frequencies, are obviously taken into account, when available.

As repeatedly pointed out by de Finetti, updating one's mind in view of new evidence does not mean changing opinion: “If we reason according to Bayes' theorem, we do not change our opinion. We keep the same opinion, yet updated to the new situation. If yesterday I was saying “It is Wednesday”, today I would say “It is Thursday”. However I have not changed my mind, for the day after Wednesday is indeed Thursday” [ de Finetti, 1995 , English edition 2008, p. 43]. In other words, the idea of correcting previous opinions is alien to his perspective, and so is the notion of a self-correcting procedure, retained by other authors, like Hans Reichenbach.

The following passage from the book Filosofia della probabilità , recently published in English under the title Philosophical Lectures on Probability , highlights de Finetti's deeply felt conviction that subjective Bayesianism is the only acceptable way of addressing probabilistic inference, and the whole of statistics. The passage also gives the flavour of de Finetti's incisive prose:

“The whole of subjectivistic statistics is based on this simple theorem of calculus of probability [Bayes' theorem]. This provides subjectivistic statistics with a very simple and general foundation. Moreover, by grounding itself on the basic probability axioms, subjectivistic statistics does not depend on those definitions of probability that would restrict its field of application (like, e.g., those based on the idea of equally probable events). Nor, for the characterization of inductive reasoning, is there any need — if we accept this framework — to resort to empirical formulae. Objectivistic statisticians, on the other hand, make copious use of empirical formulae. The necessity to resort to them only derives from their refusal to allow the use of the initial probability. […] they reject the use of the initial probability because they reject the idea that probability depends on a state of information. However, by doing so, they distort everything: not only as they turn probability into an objective thing […] but they go so far as to turn it into a theological entity: they pretend that the ‘true’ probability exists, outside ourselves, independently of a person's own judgement”. [ de Finetti, 1995 , English edition 2008, p. 43]

For de Finetti objective probability is not only useless, but meaningless, like all metaphysical notions. This attitude is epitomized by the statement “probability does not exist”, printed in capital letters in the “Preface” to the English edition of Teoria delle probabilità . A similar statement opens the article “Probabilità” in the Enciclopedia Einaudi : “Is it true that probability ‘exists’? What could it be? I would say no, it does not exist” [ de Finetti, 1980 , p. 1146]. Such aversion to the ascription of an objective meaning to probability is a direct consequence of de Finetti's anti-realism, and is inspired by the desire to keep the notion of probability free from metaphysics.

Unfortunately, de Finetti's statement has fostered the feeling that subjectivism is surrounded by a halo of arbitrariness. Against this suspicion, it must be stressed that de Finetti's attack on objective probability did not prevent him from taking seriously the issue of objectivity. In fact he struggled against the “distortion” of “identifying objectivity and objectivism”, deemed a “dangerous mirage” [ de Finetti, 1962a , p. 344], but did not deny the problem of the objectivity of probability evaluations. To clarify de Finetti's position, it is crucial to keep in mind de Finetti's distinction between the definition and the evaluation of probability. These are seen by de Finetti as utterly different concepts which should not be conflated. To his eyes, the confusion between the definition and the evaluation of probability imprints all the other interpretations of probability, namely frequentism, logicism and the classical approach. Upholders of these viewpoints look for a unique criterion — be it frequency, or symmetry — and use it as grounds for both the definition and the evaluation of probability. In so doing, they embrace a “rigid” attitude towards probability, which consists “in defining (in whatever way, according to whatever conception) the probability of an event, and in univocally determining a function” [ de Finetti, 1933 , p. 740]. By contrast, subjectivists take an “elastic” attitude, according to which the choice of one particular function is not committed to a single rule or method: “the subjective theory […] does not contend that the opinions about probability are uniquely determined and justifiable. Probability does not correspond to a self-proclaimed ‘rational’ belief, but to the effective personal belief of anyone” [ de Finetti, 1951 , p. 218]. For subjectivists there are no “correct” probability assignments , and all coherent functions are admissible. The choice of one particular function is regarded as the result of a complex and largely context-dependent procedure. To be sure, the evaluation of probability should take into account all available evidence, including frequencies and symmetries. However, it would be a mistake to put these elements, which are useful ingredients of the evaluation of probability, at the basis of its definition.

De Finetti calls attention to the fact that the evaluation of probability involves both objective and subjective elements. In his words: “Every probability evaluation essentially depends on two components: (1) the objective component, consisting of the evidence of known data and facts; and (2) the subjective component, consisting of the opinion concerning unknown facts based on known evidence' [ de Finetti, 1974 , p. 7]. The subjective component is seen as unavoidable, and for de Finetti the explicit recognition of its role is a prerequisite for the appraisal of objective elements. Subjective elements in no way “destroy the objective elements nor put them aside, but bring forth the implications that originate only after the conjunction of both objective and subjective elements at our disposal” [ de Finetti, 1973 , p. 366]. De Finetti calls attention to the fact that the collection and exploitation of factual evidence, the objective component of probability judgments, involves subjective elements of various kinds, like the judgment as to what elements are relevant to the problem under consideration, and should enter into the evaluation of probabilities. In practical situations a number of other factors influence probability evaluations, including the degree of competence of the evaluator, his optimistic or pessimistic attitudes, the influence exercised by most recent facts, and the like. Equally subjective for de Finetti is the decision on how to let belief be influenced by objective elements.

Typically, when evaluating probability one relies on information regarding frequencies. Within de Finetti's perspective, the interaction between degrees of belief and frequencies rests on exchangeability. Assuming exchangeability, whenever a considerable amount of information on frequencies is available this will strongly constrain probability assignments. But information on frequencies is often scant, and in this case the problem of how to obtain good probability evaluations becomes crucial. This problem is addressed by de Finetti in a number of writings, partly fruit of his cooperation with Savage. 47 The approach adopted is based on penalty methods , of the kind of the well known “Brier's rule”. Scoring rules like Brier's are devised to oblige those who make probability evaluations to be as accurate as they can and, if they have to compete with others, to be honest. Such rules play a twofold role within de Finetti's approach. In the first place, they offer a suitable tool for an operational definition of probability, which is in fact adopted by de Finetti in his late works. In addition, these rules offer a method for improving probability evaluations made both by a single person and by several people, because they can be employed as methods for exercising “self-control”, as well as a “comparative control” over probability evaluations [ de Finetti, 1980 , p. 1151]. 48 The use of such methods finds a simple interpretation within de Finetti's subjectivism: “though maintaining the subjectivist idea that no fact can prove or disprove belief” — he writes — “I find no difficulty in admitting that any form of comparison between probability evaluations (of myself, of other people) and actual events may be an element influencing my further judgment, of the same status as any other kind of information” [ de Finetti, 1962a , p. 360]. De Finetti's work in this connection is in tune with a widespread attitude, especially among Bayesian statisticians, that has given rise to a vast literature on “well-calibrated” estimation methods.

Having clarified that de Finetti's refusal of objective probability is not tantamount to a denial of objectivity, it should be added that such a refusal leads him to overlook notions like “chance” and “physical probability”. Having embraced the pragmatist conviction that science is just a continuation of everyday life, de Finetti never paid much attention to the use made of probability in science, and held that subjective probability can do the whole job. Only the volume Filosofia della probabilità includes a few remarks that are relevant to the point. There de Finetti admits that probability distributions belonging to scientific theories — he refers specifically to statistical mechanics — can be taken as “more solid grounds for subjective opinions” [ de Finetti, 1995 , English edition 2008, p. 63]. This allows for the conjecture that late in his life de Finetti must have entertained the idea that probabilities encountered in science derive a peculiar “robustness” from scientific theories. 49 Unlike Ramsey, however, de Finetti did not feel the need to include in his theory a notion of probability specifically devised for application in science.

With de Finetti's subjectivism, the epistemic conception of probability is committed to a theory that could not be more distant from Laplace's perspective. Unsurprisingly, de Finetti holds that “the belief that the a priori probabilities are distributed uniformly is a well defined opinion and is just as specific as the belief that these probabilities are distributed in any other perfectly specified manner” [ de Finetti, 1951 , p. 222]. But what is more important is that the weaker assumption of exchangeability allows for a more flexible inferential method than Laplace's method based on independence. Last but not least, unlike Laplace de Finetti is not a determinist. He believes that in the light of modern science, we have to admit that events are not determined with certainty, and therefore determinism is untenable. 50 For an empiricist and pragmatist like de Finetti, both determinism and indeterminism are unacceptable, when taken as physical, or even metaphysical, hypotheses; they can at best be useful ways of describing certain facts. In other words, the alternative between determinism and indeterminism “is undecidable and (I should like to say) illusory. These are metaphysical diatribes over ‘things in themselves’; science is concerned with what ‘appears to us’, and it is not strange that, in order to study these phenomena it may in some cases seem more useful to imagine them from this or that standpoint” [ de Finetti, 1976 , p. 299].

The Bayesian Decision-Theoretic Approach to Statistics

Paul Weirich , in Philosophy of Statistics , 2011

2.2 Representation Theorems

A decision problem's presentation lists options, states, and outcomes. An agent has preferences among outcomes and options. She also assigns probabilities to states and utilities to outcomes and options. How are these preferences, probabilities, and utilities related?

Representation theorems show that if a person's preferences comply with certain plausible axioms for relations among preferences, then the person's preferences entail a probability assignment to possible states of the world. Some preference axioms express requirements of rationality. For example, one axiom requires that preferences be transitive. Other axioms are structural. Compliance with them ensures a rich structure of preferences. For example, one structural axiom requires that, for any two options such that a person prefers one to the other, a third option has an intermediate preference rank. To illustrate, suppose that a person prefers betting that Heads turns up on a coin toss to betting that a six turns up on a roll of a die. Then for another proposition such as that drawing a card from a standard deck yields a diamond, she prefers betting that Heads turns up to betting that a diamond is drawn, and prefers betting that a diamond is drawn to betting that a six turns up.

The normative axioms are principles of rationality for the preferences of an ideal agent. Meeting them is necessary for having preferences that follow expected utilities. The nonnormative, structural axioms are not principles of rationality. Meeting them is not necessary for having preferences that follow expected utilities. The structural axioms ensure that preferences among options are extensive enough to derive probabilities of states from those preferences assuming that the preferences follow expected utilities.

A typical representation theorem shows that if a person's preferences concerning gambles satisfy the preference axioms, then there is a unique assignment of probabilities to possible states such that a person prefers betting that one possible state obtains to betting that another possible state obtains just in case the expected utility of the first bet exceeds the expected utility of the second bet.

Some very general representation theorems show that one may infer an agent's probabilities and utilities simultaneously. They show that if an agent's preferences concerning gambles satisfy certain axioms, then there is a unique assignment of probabilities and also a unique assignment of utilities (given a choice of scale) such that the agent prefers one gamble to another just in case the first's expected utility exceeds the second's expected utility. Savage [1954/1972] presents a famous representation theorem of this sort. His proof uses techniques that Ramsey [1931/1990] pioneered. Section 7, an appendix, presents the axioms of preference that Savage's representation theorem assumes.

A general representation theorem does not justify the laws of probability and the expected utility principle. As explained, it shows only that given any preference ordering of options that satisfies the preference axioms, there is a unique assignment of numbers to possible states of the world complying with the laws of probability, and (given a choice of scale) a unique assignment of numbers to outcomes complying with the laws of utility, such that the expected utilities of options (as computed from these probability and utility assignments) accurately reflect the preference ordering of options. The expected utility principle, for example, requires that an option's utility equal its expected utility. It is a normative principle. Its justification requires normative principles involving probabilities and utilities and not just a representation theorem's normative axioms of preference.

Evidential Probability and Objective Bayesian Epistemology

Gregory Wheeler , Jon Williamson , in Philosophy of Statistics , 2011

2.1 Motivation

Rudolf Carnap [Carnap, 1962] drew a distinction between probability 1 , which concerned rational degrees of belief, and probability 2 , which concerned statistical regularities. Although he claimed that both notions of probability were crucial to scientific inference, Carnap practically ignored probability 2 in the development of his systems of inductive logic. Evidential probability (EP) [ Kyburg, 1961 ; Kyburg and Teng, 2001 ], by contrast, is a theory that gives primacy to probability 2 , and Kyburg's philosophical program was an uncompromising approach to see how far he could go with relative frequencies. Whereas Bayesianism springs from the view that probability 1 is all the probability needed for scientific inference, EP arose from the view that probability 2 is all that we really have.

The theory of evidential probability is motivated by two basic ideas: probability assessments should be based upon relative frequencies, to the extent that we know them, and the assignment of probability to specific individuals should be determined by everything that is known about that individual. Evidential probability is conditional probability in the sense that the probability of a sentence χ is evaluated given a set of sentences Γ δ . But the evidential probability of χ given Γ δ , written Prob( χ ,Γ δ ), is a meta-linguistic operation similar in kind to the relation of provability within deductive systems.

The semantics governing the operator Prob(·,·) is markedly dissimilar to axiomatic theories of probability that take conditional probability as primitive, such as the system developed by Lester Dubbins [ Dubbins, 1975 ; Arló-Costa and Parikh, 2005 ], and it also resists reduction to linear [de Finetti, 1974] as well as lower previsions [Walley, 1991] . One difference between EP and the first two theories is that EP is interval-valued rather than point-valued, because the relative frequencies that underpin assignment of evidential probability are typically incomplete and approximate. But more generally, EP assignments may violate coherence. For example, suppose that χ and φ are sentences in the object language of evidential probability. The evidential probability of χ ∧ φ given Γ δ might fail to be less than or equal to the evidential probability that χ given Γ δ . 1 A point to stress from the start is that evidential probability is a logic of statistical probability statements, and there is nothing in the activity of observing and recording statistical regularities that guarantees that a set of statistical probability statements will comport to the axioms of probability. So, EP is neither a species of Carnapian logical probability nor a kind of Bayesian probabilistic logic. 2 , 3 EP is instead a logic for approximate reasoning, thus it is more similar in kind to the theory of rough sets [Pawlak, 1991] and to systems of fuzzy logic [Dubois and Prade, 1980] than to probabilistic logic.

The operator Prob(·,·) takes as arguments a sentence χ in the first coordinate and a set of statements Γ δ in the second. The statements in Γ δ represent a knowledge base, which includes categorical statements as well as statistical generalities. Theorems of logic and mathematics are examples of categorical statements, but so too are contingent generalities. One example of a contingent categorical statement is the ideal gas law. EP views the propositions “2 + 2 = 4” and “ PV = nRT ” within a chemistry knowledge base as indistinguishable analytic truths that are built into a particular language adopted for handling statistical statements to do with gasses. In light of EP's expansive view of analyticity, the theory represents all categorical statements as universally quantified sentences within a guarded fragment of first-order logic [Andréka et al. , 1998] . 4

Statistical generalities within Γ δ , by contrast, are viewed as direct inference statements and are represented by syntax that is unique to evidential probability. Direct inference , recall, is the probability assigned a target subclass given known frequency information about a reference population, and is often contrasted to indirect inference , which is the assignment of probability to a population given observed frequencies in a sample. Kyburg's ingenious idea was to solve the problem of indirect inference by viewing it as a form of direct inference. Since the philosophical problems concerning direct inference are much less contentious than those raised by indirect inference, the unusual properties and behavior of evidential probability should be weighed against this achievement [Levi, 2007] .

Direct inference statements are statements that record the observed frequency of items satisfying a specified reference class that also satisfy a particular target class, and take the form of

This schematic statement says that given a sequence of propositional variables x → that satisfies the reference class predicate ρ , the proportion of ρ that also satisfies the target class predicate τ is between l and u .

Syntactically, ‘ τ ( x → ), ρ ( x → ),[ l , u ]’ is an open formula schema, where ‘ τ (·)’ and ‘ ρ (·)’ are replaced by open first-order formulas, ‘ x → ’ is replaced by a sequence of propositional variables, and ‘[ l , u ]’ is replaced by a specific sub-interval of [0,1]. The binding operator ‘%’ is similar to the ordinary binding operators (∀,∃) of first-order logic, except that ‘%’ is a 3-place binding operator over the propositional variables appearing the target formula τ ( x → ) and the reference formula ρ ( x → ), and binding those formulas to an interval. 5 The language L ep of evidential probability then is a guarded first-order language augmented to include direct inference statements. There are additional formation rules for direct inference statements that are designed to block spurious inference, but we shall pass over these details of the theory. 6 An example of a direct inference statement that might appear in Γ δ is

which expresses that the proportion of As that are also B s lies between 0.71 and 0.83.

As for semantics, a model M of L ep is a pair, 〈 D I 〉 , where D is a two-sorted domain consisting of mathematical objects, D m , and a finite set of empirical objects, D e . EP assumes that there is a first giraffe and a last carbon molecule. I is an interpretation function that is the union of two partial functions, one defined on D m and the other on D e . Otherwise M behaves like a first-order model: the interpretation function I maps (empirical/mathematical) terms into the (empirical/mathematical) elements of D , monadic predicates into subsets of D , n -arity relation symbols into D n , and so forth. Variable assignments also behave as one would expect, with the only difference being the procedure for assigning truth to direct inference statements.

The basic idea behind the semantics for direct inference statements is that the statistical quantifier ‘%’ ranges over the finite empirical domain D e , not the field terms l , u that denote real numbers in D m . This means that the only free variables in a direct inference statement range over a finite domain, which will allow us to look at proportions of models in which a sentence is true. A satisfaction set of an open formula φ whose only free n variables are empirical in the subset of D n that satisfies φ .

A direct inference statement % x ( τ ( x ), ρ ( x ),[ l , u ]) is true in M under variable assignment v iff the cardinality of the satisfaction sets for the open formula ρ under v is greater than 0 and the ratio of the cardinality of satisfaction sets for τ ( x* ) ∧ ρ ( x ∗ ) over the cardinality of the satisfaction sets for ρ ( x ) (under v ) is in the closed interval [ l , u ], where all variables of x occur in ρ , all variables of τ occur in ρ , and x* is the sequence of variables free in ρ but not bound by % x [Kyburg and Teng, 2001] .

The operator Prob(·,·) then provides a semantics for a nonmonotonic consequence operator [ Wheeler, 2004 ; Kyburg et al. , 2007 ]. The structural properties enjoyed by this consequence operator are as follows: 7

Properties of EP Entailment: Let ⊨ denote classical consequence and let ≡ denote classical logical equivalence. Whenever μ ∧ ξ , ν ∧ ξ are sentences of L ep ,

Right Weakening : if μ ⊨ ν and ν ⊨ ξ then μ ⊨ ξ .

Left Classical Equivalence : if μ ⊨ ν and μ ≡ ξ then ξ ⊨ ν .

(KTW) Cautious Monotony : if μ ⊨ ν and μ ⊨ ξ then μ ∧ ξ ⊨ ν .

(KTW) Premise Disjunction : if μ ⊨ ν and ξ ⊨ ν then μ ∨ ξ ⊨ ν .

(KTW) Conclusion Conjunction : if μ ⊨ ν and μ ⊨ ξ then μ ⊨ ν ∧ ξ .

As an aside, this qualitative EP-entailment relation presents challenges in handling disjunction in the premises since the KTW disjunction property admits a novel reversal effect similar to, but distinct from, Simpson's paradox [ Kyburg et al. , 2007 ; Wheeler, 2007 ]. This raises a question over how best to axiomatize EP. One approach, which is followed by [Hawthorne and Makinson, 2007] and considered in [Kyburg et al. , 2007] , is to replace Boolean disjunction by ‘exclusive-or’. While this route ensures nice properties for ⊨, it does so at the expense of introducing a dubious connective into the object language that is neither associative nor compositional. 8 Another approach explored in [Kyburg et al. , 2007] is a weakened disjunction axiom (KTW Or) that yields a sub-System P nonmonotonic logic and preserves the standard properties of the positive Boolean connectives.

Now that we have a picture of what EP is, we turn to consider the inferential behavior of the theory. We propose to do this with a simple ball-draw experiment before considering the specifics of the theory in more detail in the next section.

EXAMPLE 1. Suppose the proportion of white balls ( W ) in an urn ( U ) is known to be within [.33,4], and that ball t is drawn from U . These facts are represented in Γ δ by the sentences, % x ( W ( x ), U ( x ),[.33,.4]) and U ( t ).

If these two statements are all that we know about t , i.e., they are the only statements in Γ δ pertaining to t , then Prob( W ( t ),Γ δ ) = [.33,.4].

the probability that ball t is white is between [.33,.4], by reason of % x ( W ( x ), U ( x ),[.33,.4]), or

the probability that ball t is white is between [.31,.36], by reason of % x ( W ( x ), P ( x ),[.31,.36]),

Table 1 . Compound Experiment

We are interested in the probability that t is white, but we have a conflict. Given these over all precise values, we would have Prob ( W ( t ) , Γ δ ) = 9 25 . However, since we know that t was selected by performing this compound experiment, then we also have the conflicting direct inference statement % x , y ( W* ( x , y ), U* ( x , y ),[.4,.4]), where U* is the set of compound two stage experiments, and W* is the set of outcomes in which the ball selected is white. 9 We should prefer the statistics from the compound experiment because they are richer in information. So, the probability that t is white is .4.

Finally, if there happens to be no statistical knowledge in Γ δ pertaining to t , then we would be completely ignorant of the probability that t is white. So in the case of total ignorance, Prob( W ( t ),Γ δ ) = [0,1].

We now turn to a more detailed account of how EP calculates probabilities.

Challenges to Bayesian Confirmation Theory

John D. Norton , in Philosophy of Statistics , 2011

4 Additivity

4.1 the property.

Probability measures are additive measures. That means that they conform to the condition:

Additivity is the most distinctive feature of the probability calculus. In the context of confirmation theory, it amounts to assigning a particular character to the degrees of belief of the theory. The degrees of this additive measure will span from a maximum value for an assured outcome (conventionally chosen as one) to a minimum value for an impossible outcome (which must be zero 20 ). If the high values are interpreted as belief with unit probability certainty, then it follows that the low values must represent disbelief, with zero probability complete disbelief. To see this, recall that near certain or certain belief in A corresponds to near complete or complete disbelief in its negation, ∼   A . 21 If we assign high probability 0.99 or unit probability to some outcome A , then by additivity the probability assigned to the negation ∼   A is 0.01 or zero, so that these low or zero values correspond to near complete disbelief or complete disbelief.

This interpretation of low probability as disbelief is already expressed in the functional dependency between the probabilities of outcomes and those of their negations. We have P (∼   A | C ) = 1 − P ( A | C ), which entails

More generally, taking relative negations, P (∼   A & B | C ) = P ( B | C ) − P ( A & B | C ), we recover the more functional dependency

and a strictly increasing function of P ( B | C ).

These dependencies tell us that a high probability assigned to an outcome corresponds to a low probability assigned to its negation or relative negation, which is the characteristic property of a scale of degrees that spans from belief to disbelief.

While additivity ( A ) entails weaker functional dependencies ( A ′) and ( A ′′), the gap between them and ( A ) is not great. If the functional dependencies ( A ′) and ( A ′′) are embedded in a natural context, it is a standard result in the literature that the resulting degrees can be rescaled to an additive measure satisfying ( A ). (See [ Aczel, 1966 , pp. 319–24; Norton, 2007 , §7].)

4.2 Non-Additive Measures: Disbelief versus Ignorance

Once it is recognized that the additivity ( A ) of the probability calculus amounts to selecting a particular interpretation of the degrees, then the ensuing challenge becomes inevitable. In many epistemic situations, we may want low degrees to represent ignorance or some mix of ignorance and disbelief, where ignorance amounts to a failure to commit to belief or, more simply, an absence of belief or disbelief. Shafer [1976, pp. 22–25] considers how we might assign degrees of belief to the proposition that there are living beings in orbit around Sirius, an issue about which we should suppose we know nothing. No additive measure is appropriate. If we assign low probability to “life,” additivity then requires us to assign high probability to “no-life.” That asserts high certainty in there being no life, something we do know. The natural intermediate of probability 1/2 for each of the two outcomes “life” and “no-life” fails to be a usable ignorance value since it cannot be used if we are ignorant over more than two mutually exclusive outcomes.

This example makes clear that introducing an element of ignorance requires a relaxation of the functional dependencies ( A ′) and ( A ′′). It must be possible to assign a low degree of belief to some outcome without being thereby forced to assign a high degree to its negation. Measures that allow this sort of violation of additivity are “superadditive”: if A and B are mutually exclusive then, for any C , P ( A ∨ B | C ) ≥ P ( A | C ) + P ( B | C ). The extent to which the equality is replaced by an inequality is the extent to which the measure allows representation of ignorance.

The best-known superadditive calculus is the Shafer-Dempster calculus. In it, an additive measure m , called a “basic probability assignment ,” is defined over the power set of the “frame of discernment” Θ, so that Σ A ⊆Θ m ( A ) = 1, where the summation extends overall all subsets of Θ, and m({}) = 0. This basic measure probability assignment is used to generate the quantity of interest, the “belief function” Bel, which is defined as

for any A in Θ, where the summation is taken over all subsets B of A . These belief functions allow blending of disbelief and ignorance. For example, that we are largely ignorant over the truth of “life” can be represented by

which induces the belief function

4.3 Complete Ignorance: The First Problem of the Priors

In Bayesian confirmation theory, prior probability distributions are adjusted by Bayes' theorem to posterior probability distributions that incorporate new evidence learned. As we trace this chain back, the prior probability distributions represent states of greater ignorance. Bayesian confirmation theory can only give a complete account of this learning process if it admits an initial state representing complete ignorance. However representing ignorance is a long-standing difficulty for Bayesian confirmation theory and the case of complete ignorance has been especially recalcitrant. It is designated here as the first problem of the priors, to distinguish it from another problem with prior probability delineated below in Section 5.6 below.

There are two instruments already in the probability literature that are able to delimit the representation of this extreme case, the epistemic state of complete ignorance. Norton [2008] has given an extended analysis of how the two may be used to do this. The first instrument is the principle of indifference. It asserts that if we have no grounds for preferring one outcome to a second, then we should assign equal belief to both. This platitude of evidence is routinely used to ground the classical interpretation of probability and famously runs into trouble when we redescribe the outcome space. Complete ignorance in one description is equivalent to complete ignorance in another. That fact allows one to infer quite rapidly that the degree of belief assigned to some compound proposition A ∨ B should be the same as the degree of belief assigned to each of the disjunctive parts A and B , even though A and B may be mutually exclusively. 22 Since no probability distribution can have this property, it is generally concluded that there is something wrong with the principle of indifference.

The difficulty is that the principle of indifference is not so easily discarded. It is a platitude of evidence. If beliefs are grounded in reasons and we have no reasons to distinguish two outcomes, then we should have the same belief in each. The alternative is to retain the principle of indifference and discard the notion that a probability distribution can adequately represent complete ignorance. Instead we are led to a representation of complete ignorance by a non-probabilistic distribution with three values: Max and Min for the extreme values of certainty and complete disbelief and Ig (“ignorance”) for everything in between

The essential property here is that we can assign the ignorance degree Ig to some contingent outcome A ∨ B and that same ignorance degree to each of its mutually exclusive, disjunctive parts, A and B . This is the only distribution for which this is true over all contingent outcomes. 23

The second instrument used to delineate the epistemic state of complete ignorance is the notion of invariance, used so effectively by objective Bayesians, but here used in a way that objective Bayesians may not endorse. The notion ultimately produces serious problems for Bayesian confirmation theory. The greater our ignorance, the more symmetries we have under which the epistemic state should be invariant. It is quite easy to accumulate so many symmetries that the epistemic state cannot be a probability distribution. For example, let us say we know only

This information (DATUM) remains true if x is replaced by x ′ = 1 − x , where the function x ′( x ) is self-inverting. It also remains true if x is replaced by x ′′ = 1−(1−(1− x ) 2 ) 1/2 , where once again the function x” (x) is self-inverting. So our epistemic state must remain unchanged under each of these transformations. It is easy to show that no probability distribution can be invariant under both. (See [Norton, 2008 , §3.2].)

Invariance requirements can be used to pick out the unique state of complete ignorance, which turns out to be (I) above. To see the relevant invariance, consider some proposition A over whose truth we may be completely ignorant. Our belief would be unchanged were A replaced by ∼   A . 24 That is, the state of complete ignorance remains unchanged under a transformation that replaces every contingent proposition with its negation. It can readily be seen that the ignorance distribution (I) satisfies this invariance requirement. Each contingent proposition and its negation are assigned the same degree Ig . 25 That this ignorance distribution (I) is the only distribution satisfying this invariance that we are likely to encounter is made more precise by the demonstration that it is the only monotonic 26 distribution of belief with the requisite invariance [Norton, 2008 , §6].

The negation map — the transformation that replaces each contingent proposition with its negation — is a little more complicated than it may initially seem. To see the difficulty, imagine that the outcome space is exhausted by n mutually exclusive atomic propositions, A 1 , A 2 , …   , A n . The transformation replaces atomic propositions, such as A 1 by compound propositions, such as A 2 ∨   A 3 ∨…∨ A n , so it may not be evident that the transformation is a symmetry of the outcome space. It is, in the sense that it maps the outcome space back to itself and is self-inverting; A 2 ∨   A 3 ∨…∨ A n is mapped to A 1 . However, it does not preserve additive measures. The transformation takes an additive measure m to what Norton [2007a] describes as a “dual additive measure” M . These dual additive measures have properties that are, on first acquaintance, odd looking. They are additive, but their additivity is attached to conjunctions. If we have propositions A and B such that A ∨ B is always true, then we can add their measures as M ( A & B ) = M ( A ) + M ( B ). The notion of a dual measure allows a simple characterization of the ignorance distribution (I): it is the unique, monotonic measure that is self-dual.

If one does not see that an epistemic state of complete ignorance is represented by the non-probabilistic (I), one is susceptible to the “inductive disjunctive fallacy” [Norton, forthcoming a]. Let a 1 , a 2 , a 3 , … be a large number of mutually exclusive outcomes over which we are in complete ignorance. According to (I), we remain in that state of complete ignorance for any contingent disjunction of these outcomes, a 1 ∨   a 2 ∨   a 3 ∨… If one applies probabilities thoughtlessly, one might try to represent the state of complete ignorance by a broadly spread probability distribution over the outcomes. Then the probability of the disjunction can be brought close to unity merely by adding more outcomes. Hence one would infer fallaciously to near certainty for a sufficiently large contingent disjunction of outcomes over which we are individually in complete ignorance. The fallacy is surprisingly widespread. A striking example is supplied by van Inwagen [1996] in answer to the cosmic question “Why is there anything at all?” There is, he asserts, one way for no thing to be, but infinitely many ways for different things to be. Distributing probabilities over these outcomes fairly uniformly, we infer that the disjunction representing the infinitely many ways things can be must attract all the probability mass so that we assign probability one to it.

4.4 Bayesian Responses

The literature in Bayesian confirmation theory has long grappled with this problem of representing ignorance. That is especially so for prior probability distributions, where the presumption that ignorance must be representable is most pressing. Perhaps the most satisfactory response comes through the basic supposition of subjective Bayesians that the probabilities are subjective and may vary from person to person as long as the axioms of the probability calculus are respected. So the necessary deviation from the non-probabilistic ignorance distribution (I) in some agent's prior probability distribution is discounted as an individual aberration not reflecting the true evidential situation. The price paid in adopting this response, the injection of subjectivity and necessity of every prior being aberrant, is too high a price for objective Bayesians, who are committed to there being one probability distribution appropriate for each circumstance.

However objective Bayesian methods have not proven able to deliver true “ignorance priors,” even though the term does appear over-optimistically in the objective Bayesian literature [ Jaynes, 2003 , Ch.12]. One approach is to identify the ignorance priors by invariance properties. That meets only with limited success, since greater ignorance generates more invariances and, as we saw in Section 4.3 above, eventually there are so many invariances that no probability measure is admissible. An alternative approach is to seek ignorance priors in distributions of maximum entropy. 27 Maximum entropy distributions do supply what are, in an intuitive sense, the most uniform distributions admissible. If, for example, we have an outcome space comprising n atomic propositions, without further constraints, the maximum entropy distribution is the one uniform distribution that assigns probability 1/n to each atomic proposition. However, if there is sufficient ignorance, there will be invariances under which the property of having attainted maximum entropy will not be preserved. In the end, it is inevitable that these methods cannot deliver the ignorance distribution (I), for (I) is not a probability distribution. So the best that can be expected is that they will deliver a distribution that captures ignorance over one aspect of the problem, but not all. The tendency in the literature now is to replace the misleading terminology of “ignorance prior” by more neutral terms such as “noninformative priors,” “reference priors” or, most clearly “priors constructed by some formal rule” [ Kass and Wasserman, 1996 ].

Another popular approach to representing ignorance with Bayesian confirmation theory is to allow that an agent's epistemic state is not given by any one probability measure, but by a set of them, possibly convex. (See for example [ Levi, 1980 , Ch. 9].) The deepest concern with this strategy is that it amounts to an attempt to simulate the failure of additivity that is associated with the representation of ignorance. Something like the complete ignorance state (I) can be simulated, for example, by taking the set of all probability measures over the same outcome space. The resulting structure is vastly more complicated than (I), the state it tries to simulate. It has become non-local in the sense that a single ignorance value is no longer attributed to an outcome. Each of the many probabilities assigned to some outcome must be interpreted in the context of the other values assigned to other propositions and in cognizance that there are many other distributions in the set. 28

Finally, as pointed out in Norton [2007a; 2008] , a set of probability measures necessarily falls short of simulating (I). For no set of additive measures can have the requisite invariance property of a complete ignorance state — invariance under the negation map. For a set of additive measures is transformed by the negation map into a set of dual additive measures. In informal terms, any set of additive measures on an outcome space preserves a directedness. For each measure in the set, as one proceeds from the assuredly false proposition to the assuredly true by taking disjunctions, the measures assigned are non-decreasing and must, at some point, increase strictly. Invariance under the negation map precludes such directedness.

4.5 Ignorance over a Countable Infinity of Outcomes

The difficulties of representing ignorance have been explored in the literature in some detail in the particular problem of identifying a state of ignorance over a countable infinity of outcomes. It has driven Bayesians to some extreme proposals none of which appear able to handle the problem in its most intractable form. 29

The traditional starting place — already “well-known” when de Finetti [1972, p.86] outlined it- is to determine our beliefs concerning a natural number “chosen at random.” Its more figurative version is “de Finetti's Lottery” [ Bartha, 2004 ] in which a lottery ticket is picked at random from a countable infinity of tickets. If we write the prior probability for numbers 1, 2, 3, … as p 1 , p 2 , p 3 , …, we cannot reconcile two conditions. First, since we have no preference for any number over any other, we assign the same probability to each number

Second, the sum of all probabilities must be unity

No set of values for p i can satisfy both conditions. 30

De Finetti's own solution was to note that the condition (CA), “countable additivity” (as applied to this example), is logically stronger than finite additivity (A). The latter applies only to a finite set of outcomes — say, that the number chosen is a number in {1, 2, …, n }. It asserts that the probability of this finite set is the finite sum p 1 +   p 2 +   p 3 +…+ p n . Condition (CA) adds the requirement that this relation continues to hold in the limit of n→ ∞. De Finetti asserted that the probability distribution representing our epistemic state in this problem need only be finitely additive, not countably additive. That allows us to set p i = 0 for all i , without forcing the probability of infinite sets of outcomes to be zero. So if odd = {1, 3, 5, …} and even = {2, 4, 6, …} we can still set P ( odd ) = P ( even ) = 0.5.

Solving the problem by dropping countable additivity has proven to be a popular and well-understood solution. While proponents of the restriction to finite additivity are typically not frequentists (who identify probabilities with relative frequencies), the connection to frequentism is natural. The frequency of any particular natural number among all is zero and the frequency of even numbers is 1/2, in a naturally defined limit. Kadane and O'Hagan [1995] have mapped out which uniform, finitely additive probability distributions are possible over the natural numbers, noting how these are delimited by natural conditions such as agreement with limiting frequencies and invariance of probability under translation of a set of numbers. There are also variant forms of the proposal, such as the use of Popper functions and the notion of “relative probability” [ Bartha, and Johns, 2001, Bartha, 2004 ]. However dropping countable additivity is not without disadvantages and enthusiasm for it is not universal. There are Dutch book arguments that favor countable additivity; and important limit theorems, including Bayesian convergence of opinion theorems, depend upon countable additivity. See [ Williamson, 1999; Howson and Urbach, 2006 , pp. 26–29; Kelly, 1996 , Ch. 13]. 31 Other ap-proaches explore the possibility of assigning infinitesimally small probabilities to what would otherwise be zero probability outcomes [ McGee, 1994 ].

While all these solutions come at some cost, they are eventually unavailing. For they have not addressed the problem in its most acute form. They deal only with the case of ignorance over natural numbers, where this set of a countable infinity of outcomes has a natural order. If we presume that we know of no such natural order, then all these solutions fail, as has been shown in a paradox reported by Bartha [2004 , §5.1] and in the work of Arntzenius [manuscript].

Imagine that we have a countable infinity of outcomes with no way to order them. If we have some labeling of the outcomes by natural numbers, that numbering is completely arbitrary. 32 Let us pick some arbitrary labeling of the outcomes, 1, 2, 3, …, and seek an ignorance distribution over these labels. That ignorance distribution should be unaffected by any one-to-one relabeling of the outcomes; that is, the ignorance distribution is invariant under a permutation of the labels, for permutations are a symmetry of this system. Consider the outcomes odd = {1, 3, 5, …} and even = {2, 4, 6, …}. There is a permutation that simply switches the labels of the two sets (1 ↔ 2,3 ↔ 4,5 ↔ 6,…), so that outcomes in each set are exchanged. Since our belief distribution is invariant under such a permutation, it follows that the permutation does not alter our belief and we must have the same belief in each outcome set odd and even . Now consider the four outcome sets

There is a permutation that switches labels of one and two ; so we have equal belief in one and two . Proceeding pairwise through the sets, we find we must have equal belief in each of one , two , three and four . Now there is also a pairwise permutation that switches the labels of one with those of two ∪ three ∪ four , where

It is just the obvious permutation read off the above sets

So we now infer that we must have the same belief in one as in two ∪ three ∪ four . Combining we find: we must have the same belief in two ∪ three ∪ four and in each of its disjunctive parts two , three and four . This requirement cannot be met by a probability distribution for it contradicts (finite) additivity. 33 It is however compatible with the ignorance distribution (I).

Finally, it is sometimes remarked that the very idea of uniform ignorance over a countable set is somehow illicit for there is no mechanical contrivance that could select outcomes so that they have equal chances. No lottery commission could build a device that would implement the de Finetti lottery. For references to these concerns, see [ Bartha, 2004 , pp. 304–305], who correctly objects that the inevitable non-unformity of probabilities of the lottery machine does not force non-uniformity of beliefs. What needs to be added is that the entire objection is based on a circularity. In it, the notion of a mechanical contrivance tacitly supposes a contrivance whose outcomes are governed by a probability distribution. So it amounts to saying that no machine whose outcomes are governed by a probability distribution can generate outcomes governed by a non-probabilistic distribution. Recent work in philosophy of physics has identified idealized physical mechanisms that produce indeterministic outcomes that are not governed by a probability distribution. An example is the “dome,” described in [Norton, 2007 , §8.3; forthcoming], where it is shown that the dome's outcomes are governed by a non-probabilistic distribution with the same structure as (I). There are many more examples of these sorts of indeterministic systems in the “supertask” literature. (For a survey, see [Laraudogoitia, 2004] .) Many of these mechanisms conform with Newtonian mechanics, but depend on idealizations some find “unphysical” for their distinctive behavior. 34 These processes could be the physical basis of an idealized contrivance that implements something like the de Finetti lottery.

Related terms:

• Probability Theory
• Bayesian Inference
• Prior Probability
• Probability Function
• Statistical Hypothesis
• Proposition

Browse Course Material

Course info.

• Prof. Scott Sheffield

Departments

• Mathematics

As Taught In

• Discrete Mathematics
• Probability and Statistics

Learning Resource Types

Probability and random variables, assignments.

Assignments include problems from the required textbook.

Ross, Sheldon. A First Course in Probability . 8th ed. Pearson Prentice Hall, 2009. ISBN: 9780136033134.

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