De Finetti s ultimate failure. Krzysztof Burdzy University of Washington
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1 De Finetti s ultimate failure Krzysztof Burdzy University of Washington
2 Does philosophy matter? Global temperatures will rise by 1 degree in 20 years with probability 80%.
3 Reading suggestions Probability is Symmetry (KB) Philosophy/book.pdf D.A. Gillies Philosophical Theories of Probability Routledge, London, 2000
4 Classical statistics Frequency philosophy Bayesian statistics Subjective philosophy
5 Four mature philosophies Created in twentieth century, randomly ordered Name Logical Principal philosopher Rudolf Carnap What is the nature of probability? Weak implication Propensity Karl Popper Physical property Frequency Subjective Richard von Mises Bruno de Finetti Attribute of a sequence Personal opinion
6 Twin philosophies The frequency and subjective philosophies are the only two philosophies which claim that events do not have probabilities.
7 Probability does not exist (de Finetti) The subjective philosophy is a religion its dogmas have to be interpreted.
8 Decisions and probability De Finetti s most scientific and least controversial claim was that the rules of Bayesian inference can be derived from a system of axioms for rational decision making that did not presuppose existence of probability.
9 Lindley s challenge The talk is in part an answer to the following challenge posed by D. Lindley in his Math review of M.H. DeGroot s Optimal Statistical Decisions : Many statisticians and decision-theorists will be out of sympathy with the book because it is openly Bayesian.... But they would do well to consider the argument dispassionately and consider whether the axioms are acceptable to them. If they are, then the course is clear; if not, then they should say why and then develop their own and the deductions from them.
10 From decisions to probability De Finetti s idea has (at least) two representations. The Dutch book argument (popular with philosophers). The von Neumann-Morgenstern-Savage system of axioms (formal mathematics).
11 Axioms Decision A is strictly preferable to decision B and decision B is strictly preferable to decision A is irrational. Highest level of abstraction Elimination of irrational decision strategies No indication of how to order rational strategies Real axioms are more complicated Real people do not follow the axioms
12 Existence theorem Theorem 1 (de Finetti- von Neumann- Morgenstern-Savage). If a decision strategy is rational then there exist a probability P and a utility function U such that decision A is preferable to decision B if and only if E U(A) > E U(B).
13 Bayesian statistics Prior distribution Data Bayes theorem Posterior distribution
14 De Finetti s ultimate failure Theorem 1 can be used to show that Prior Data Prior must be a probability distribution. Bayes theorem Posterior Posterior must be a probability distribution.
15 De Finetti s ultimate failure Theorem 1 can be used to show that Prior Data ( Ω, F, P ) Bayes theorem Posterior ( Ω, F, P )
16 Investment strategy (an example) Monday: Buy stocks or bonds Tuesday: Read newspaper (new data) Wednesday: Buy stocks or bonds
17 The axioms can be applied to actions taken on Wednesday: Monday: Stocks are strictly better than bonds and bonds are strictly better than stocks is irrational.
18 None of the following investment strategies is irrational. Day Monday Wednesday Preferred investment Stocks Stocks Bonds Bonds Stocks Bonds Stocks Bonds
19 Separation of decision strategies The axioms do not specify any relationship between actions taken on Monday and actions taken on Wednesday. Theorem 1 splits in the Bayesian context into two separate theorems one on the prior actions and the other on the posterior actions.
20 Irreparable error Theorem 2 (KB). Suppose strategy for Monday is consistent and strategy S 2 for Wednesday is consistent. Then there exists a probability measure P representing both S and S 1 2 as a single case of Bayesian inference. Proof. Use Theorem 1 to find P 1 representing and representing. Let P = P P 1. S P S Data = extremely unlikely catastrophic event S 1
21 Irreparable error You cannot add an axiom relating Monday actions and Wednesday actions because that would eliminate some pairs of self-consistent ( S, ) 1 S 2 cannot do that because every pair of self-consistent strategies is a Bayesian strategy. strategies. Theorem 2 shows that you
22 Consequences of de Finetti s error For Bayesian statistics: none. For environment: horrendous. Admissibility? Bayesian inference is an excellent method of determining exact or approximate values of objective probabilities.
23 Frequency philosophy The von Mises theory can be represented as two scientific ideas. Both were totally rejected.
24 Collectives X k Definition: A collective is a sequence of 0-1 random variables such that for some and every strictly increasing sequence of predictable stopping times lim n 1 n n k = 1 X T k = p, T n a. s. p Challenge: Prove CLT for collectives. I.I.D. Collective / I.I.D.
25 Collectives and only collectives Von Mises: Probability theory can be applied only to collectives. What should we do with the data on financial markets, climate and weather, social networks, etc.?
26 Science of probability (KB) (L1) Probabilities are numbers between 0 and 1, assigned to events whose outcome may be unknown. (L2) If events A and B cannot happen at the same time then A P ( A B) = P( A) + P( B) (L3) If events and are physically independent then B P ( A B) = P( A) P( B) (L4) If there exists a symmetry on the space of possible outcomes which maps an event A onto an event B then P ( A) = P( B) (L5) An event has probability 1 if and only if it must occur.
27 Popper s view of science (L5) An event has probability 1 if and only if it must occur. (L5) is Popper s idea of falsification of probability statements (and scientific statements in general), repackaged for the mass market.
28 buy bonds on Wednesday There is no future Decisions (actions): buy stocks on Monday and buy bonds on Wednesday cannot be ordered (compared). Monday Tuesday Wednesday Physical time Probabilistic time F G
29 De Finetti: Our point of view remains in all cases the same: to show that there are rather profound psychological reasons which make the exact or approximate agreement that is observed between the opinions of different individuals very natural, but there are no reasons, rational, positive, or metaphysical, that can give this fact any meaning beyond that of a simple agreement of subjective opinions.
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