An Introduction to Bayesian Reasoning

Transcription

1 Tilburg Center for Logic and Philosophy of Science (TiLPS) Tilburg University, The Netherlands EPS Seminar, TiLPS, 9 October 2013

2 Overview of the Tutorial This tutorial aims at giving you an idea of why degrees of belief are an important concept in philosophy; why probability is a good tool for modeling degrees of belief; what kind of nice things one can do with it.

3 1 A la recherche de la croyance partielle 2 Probabilities as Degrees of Belief The Mathematics of Probability The Subjective Interpretation 3 Intermezzo: The Monty Hall problem 4 Success stories

4 Motivation 1: Epistemology and the Lottery Paradox Consider a fair 1000 ticket lottery that has exactly one winning ticket.

5 Motivation 1: Epistemology and the Lottery Paradox Consider a fair 1000 ticket lottery that has exactly one winning ticket. I am justified to believe that ticket #1 (#2, #3,... ) of the lottery will not win.

6 Motivation 1: Epistemology and the Lottery Paradox Consider a fair 1000 ticket lottery that has exactly one winning ticket. I am justified to believe that ticket #1 (#2, #3,... ) of the lottery will not win. Thus, I am justified to believe that no ticket will win (by closure of justified belief under known logical implication).

7 Motivation 1: Epistemology and the Lottery Paradox The lottery paradox was designed to demonstrate that three attractive principles governing justified belief lead to contradiction, namely that

8 Motivation 1: Epistemology and the Lottery Paradox The lottery paradox was designed to demonstrate that three attractive principles governing justified belief lead to contradiction, namely that 1 X is justified to believe a proposition that is very likely true;

9 Motivation 1: Epistemology and the Lottery Paradox The lottery paradox was designed to demonstrate that three attractive principles governing justified belief lead to contradiction, namely that 1 X is justified to believe a proposition that is very likely true; 2 X is never justified to believe a proposition that s/he knows to be inconsistent;

10 Motivation 1: Epistemology and the Lottery Paradox The lottery paradox was designed to demonstrate that three attractive principles governing justified belief lead to contradiction, namely that 1 X is justified to believe a proposition that is very likely true; 2 X is never justified to believe a proposition that s/he knows to be inconsistent; 3 If X believes A, and s/he knows that A A, then X should also believe A.

11 Motivation 1: Epistemology and the Lottery Paradox The lottery paradox was designed to demonstrate that three attractive principles governing justified belief lead to contradiction, namely that 1 X is justified to believe a proposition that is very likely true; 2 X is never justified to believe a proposition that s/he knows to be inconsistent; 3 If X believes A, and s/he knows that A A, then X should also believe A.

12 Motivation 1: Epistemology and the Lottery Paradox The lottery paradox was designed to demonstrate that three attractive principles governing justified belief lead to contradiction, namely that 1 X is justified to believe a proposition that is very likely true; 2 X is never justified to believe a proposition that s/he knows to be inconsistent; 3 If X believes A, and s/he knows that A A, then X should also believe A. Solution Proposal: The trouble may be with the first principle and the notion of full belief. Replace the latter by a full-fledged theory of partial belief (or degrees of belief).

13 Motivation 2: Explaining Action We often observe that social norms crumble with decreasing mutual trust of the participants. Beliefs about the actions of the rest of the group are important for explaining behavior in such situations. A theory of full belief seems to be incapable of modeling the gradual establishment and decline of social equilibria. Solution Proposal: partial beliefs + preference ordering over outcomes.

14 Motivation 3: Scientific Theory Confirmation In the good old days of Viennese positivism, scientific theories ( all ravens are black ) were confirmed by their instances (a black raven).

15 Motivation 3: Scientific Theory Confirmation In the good old days of Viennese positivism, scientific theories ( all ravens are black ) were confirmed by their instances (a black raven). The Raven Paradox (Hempel 1945): 1 It seems reasonable that if E confirms H, and H H, then E also confirms H.

16 Motivation 3: Scientific Theory Confirmation In the good old days of Viennese positivism, scientific theories ( all ravens are black ) were confirmed by their instances (a black raven). The Raven Paradox (Hempel 1945): 1 It seems reasonable that if E confirms H, and H H, then E also confirms H. 2 H = x : Rx Bx is logically equivalent to H = x : Bx Rx.

17 Motivation 3: Scientific Theory Confirmation In the good old days of Viennese positivism, scientific theories ( all ravens are black ) were confirmed by their instances (a black raven). The Raven Paradox (Hempel 1945): 1 It seems reasonable that if E confirms H, and H H, then E also confirms H. 2 H = x : Rx Bx is logically equivalent to H = x : Bx Rx. 3 Thus, E = Ba. Ra ( a white shoe ) confirms H and (by 1.) also H.

18 Motivation 3: Scientific Theory Confirmation Solution Proposal: Perhaps we can show that E=Ra.Ba confirms H to a higher degree than E = Ba. Ra. That is, we would need degrees of confirmation. But what do they mean?

19 Motivation 3: Scientific Theory Confirmation Solution Proposal: Perhaps we can show that E=Ra.Ba confirms H to a higher degree than E = Ba. Ra. That is, we would need degrees of confirmation. But what do they mean? The arguably ingenious move of Bayesian Confirmation Theory is to combine degrees of belief and degrees of confirmation with the help of only a single concept subjective probability.

20 The Mathematics of Probability No use for philosophers? Are formal methods at all useful for philosophers? Here is what the influential blog writer Brian Leiter claims: I ll soon make use of my classic An anonymous philosopher writes: Brian, thanks for suffering for all us and for keeping us safe. You give so much and ask for so little in return. You are the wind beneath my wings and with that all of your methodological gobbledygook will be forgotten because real philosophers shouldn t be expected to care about such things any more than they should have to know something about boolean algebras. (Leiter 2012)

21 The Mathematics of Probability Boolean Algebras Definition: A Boolean σ-algebra A over a set of propositions Ω is a subset of the power set of Ω such that 1 A; 2 A A A A; 3 A k A k N k N A k A.

22 The Mathematics of Probability Probability Definition (sentential version): A probability is a function p( ) from a Boolean σ-algebra (Ω, A) to the interval [0,1] such that 1 p( ) = 0; 2 p( A) = 1 p(a); 3 for mutually exclusive propositions A k, p( k N A k) = k N p(a k).

23 The Mathematics of Probability σ-additivity The requirement that for mutually exclusive propositions A k, p( k N A k) = k N p(a k), is known as σ-addivitity. Mathematical Reason Relation to measure theory. Allows for countable approximations, e.g. in the computation of integrals.

24 The Mathematics of Probability σ-additivity The requirement that for mutually exclusive propositions A k, p( k N A k) = k N p(a k), is known as σ-addivitity. Mathematical Reason Relation to measure theory. Allows for countable approximations, e.g. in the computation of integrals. Conceptual Reason Allows to build probability logic over languages with countably many predicates. Denying it leads to some strange consequences, e.g., failure of conglomerability.

25 The Subjective Interpretation Probability as subjective degree of belief Figure: The Reverend Thomas Bayes (1763) proposed the interpretation of probability as subjective degree of belief.

26 The Subjective Interpretation Belief and bets (I) Speed: Sir, Proteus, save you! Saw you my master? Proteus: But now he parted hence, to embark for Milan. Speed: Twenty to one, then, that he s shipped already! (Shakespeare, Two Gentlemen of Verona )

27 The Subjective Interpretation Beliefs and bets (II) Ramsey (1926) proposed to operationalize degrees of belief in terms of betting odds. [... ] all our lives we are in sense betting. Whenever we go to the station we are betting that a train will really run, and if we had not a sufficient degree of belief in this we should decline the bet and stay at home. (Ramsey [1926] 1978, 85.)

28 The Subjective Interpretation The Dutch Book Argument Why should degrees conform to the probability calculus?

29 The Subjective Interpretation The Dutch Book Argument Why should degrees conform to the probability calculus? Because this is the only way we don t lose money if we operationalize degrees of belief in terms of betting behavior. A synchronously held set of beliefs over all propositions of our Boolean algebra must conform to the probability calculus. Otherwise a cunning bookie will exploit us by means of a Dutch Book, that is, a sure loss system of bets.

30 The Subjective Interpretation The Dutch Book Argument (cont d) Dutch Book Theorem: Any system of betting odds over a Boolean σ-algebra that does not satisfy the axioms of probability is vulnerable to a Dutch Book.

31 The Subjective Interpretation The Dutch Book Argument (cont d) Dutch Book Theorem: Any system of betting odds over a Boolean σ-algebra that does not satisfy the axioms of probability is vulnerable to a Dutch Book. Converse Dutch Book Theorem (De Finetti 1937, Kemeny 1955): A system of betting odds over a Boolean σ-algebra that satisfies the axioms of probability is not vulnerable to a Dutch Book.

32 The Subjective Interpretation Learning from Experience Degrees of belief are changed according to Bayesian Conditionalization: p e (h) = p(h e)

33 The Subjective Interpretation Learning from Experience Degrees of belief are changed according to Bayesian Conditionalization: p e (h) = p(h e) This conditional probability is calculated by means of a simple fact of probability theory, Bayes s Theorem (Bayes 1763): p(h e) = p(h) p(e h) p(e) ( = 1 + p(e h) p(e h) p( h) p(h) ) 1

34 The Subjective Interpretation Conditionalization and Induction Question: Is Bayesian updating a satisfactory reply to the problem of induction? quasi-analytic, normatively compelling belief revision? logic or justification of inductive inference? where do the prior probabilities come from?

35 The Monty Hall problem Suppose you re on a game show, and you re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what s behind the doors, opens another door [but never the door that hides a prize], say No. 2, which has a goat. He then says to you, Do you want to pick door No. 3? Is it to your advantage to switch your choice?

36 A Bayesian analysis of Monty Hall The confusion emerges because various conceptions of probability (the classical one and the Bayesian one) interfere. There are various arguments that elucidate why switching is rational: Compare the long-run success frequencies of switching vs. sticking with your choice. Analogy to a case of doors, all of which, apart from two doors, are opened. The opening of door 2 favors the hypothesis that the car is behind door No. 3 over the hypothesis that it is behind door No. 1.

37 A Bayesian analysis of Monty Hall Let D k be the proposition that door k is the winning one, and let E be the evidence that door 2 is opened and reveals a goat. Then we obtain by means of Bayesian updating p(d 3 E) = p(d 3 ) p(e D 3) p(e) = ( 1 + p(e D 3) p(e D 3 ) p( D ) 1 3) p(d 3 ) The a-priori probabilities are p(d 3 ) = 1/3 and therefore p( D 3 ) = 2/3. p(e D 3 ) = 1 (because Monty could not have opened a different door). Finally, p(e D 3 ) = 1/2(p(E D 1 ) + p(e D 2 )) = 1/4. This implies that our posterior probability p(d 3 E) takes the value 2/3!

38 The Lottery Paradox Degrees of belief allow for a straightforward analysis of the lottery paradox: From the perspective of partial belief, the second and third principle are vindicated: Never believe inconsistent propositions (probability zero). Logical consequences of a proposition A should believed as least as strongly as A. The problem is with the first principle X is justified to believe a proposition that is very likely true. This is false even high partial belief does not warrant justified belief, rational acceptance or the like.

39 Explaining Behavior: Representation Theorems Typical Representation Theorem (van Neumann-Morgenstern, Savage): If the preferences of an agent X are transitive, complete, reflexive,..., then they can be represented as emerging from a unique probability function p and a real-valued utility function u (unique up to affine transformation) on the set of all outcomes.

40 Explaining Behavior: Representation Theorems Typical Representation Theorem (van Neumann-Morgenstern, Savage): If the preferences of an agent X are transitive, complete, reflexive,..., then they can be represented as emerging from a unique probability function p and a real-valued utility function u (unique up to affine transformation) on the set of all outcomes. Theorem (Savage, 1954): If the preference relation satisfies the Savage postulates P1-P7, then (and only then) there exists a unique, nonatomic, finitely additive probability measure p on S, and a bounded, unique up to a positive affine transformation real-valued function u such that for all acts f and g, f g if and only if u(f (S))dp(s) u(g(s)) dp(s) S S

41 Confirmation and the Raven Paradox One can show that under very plausible and weak assumptions, the observation E = Ra.Ba confirms H = x : Rx Bx to a higher degree than E = Ba. Ra. (Hawthorne and Fitelson 2011) There is a wide variety of probabilistic measures for degree of confirmation that are, partly, also used in statistical inference: l(h, e) = p(e h) p(e h) r(h, e) = p(h e) p(h) d(h, e) = p(h e) p(h) z(h, e) = { p(h e) p(h) 1 p(h) p(h e) p(h) p(h) if p(h e) > p(h) if p(h e) < p(h)

42 Further applications and challenges Some classic problems in philosophy of science can also be approached from a Bayesian perspective: Duhem-Quine Problem Problem of Old Evidence Logic of Explanatory Reasoning...

43 Thanks a lot for your attention! This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.