Poirier-Handbook+Final - Exchangeability Representation...

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Exchangeability, Representation Theorems, and Subjectivity Dale J. Poirier University of California, Irvine January 1, 2010 Abstract According to Bruno de Finetti’s Representation Theorem, exchangeable beliefs over infinite sequences of observable Bernoulli quantities can be represented as mixtures of independent coin tossing experiments. Extensions of this theorem give rise to representations as mixtures of other familiar sampling distributions. This paper offers a subjectivist primer based on the premise that appreciation of these theorems enhances understanding of the subjective interpretation of probability, of its connection to the more prevalent frequentist interpretation, and of its usefulness as a context in which to view parameters, priors, and likelihoods. 1. Introduction In his popular notes on the theory of choice, (Kreps, 1988, p. 145) opines that Bruno de Finetti’s Representation Theorem is the fundamental theorem of statistical inference. de Finetti’s theorem characterizes likelihood functions in terms of symmetries and invariance. The conceptual framework begins with an infinite string of observable random quantities taking on values in a sample space Z . Then it postulates symmetries and invariance for probabilistic assignments to such strings, and finds a likelihood with the prescribed properties for finite strings of length N. This
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2 theorem, and its generalizations: (i) provide tight connections between Bayesian and frequentist reasoning, (ii) endogenize the choice of likelihood functions, (iii) prove the existence of priors, (iv) provide an interpretation of parameters which differs from that usually considered, (v) produce Bayes’ Theorem as a corollary, (vi) produce the Likelihood Principle (LP) and Stopping Rule Principle (SRP) as corollaries, and (vii) provide a solution to Hume’s problem of induction. This is a large number of results. Surprisingly, these theorems are rarely discussed in econometrics. de Finetti developed subjective probability during the 1920s independently of (Ramsey, 1926). de Finetti was an ardent subjectivist. He is famous for the aphorism: “Probability does not exist.” By this he meant that probability reflects an individual’s beliefs about reality, rather than a property of reality itself. This viewpoint is also “objective” in the sense of being operationally measurable, e.g., by means of betting behavior or scoring rules. For example, suppose your true subjective probability of some event A is p and the scoring rule is quadratic [ 1 (A) - ] 2 , where 1 (A) is the indicator function and is your announced probability of A occurring. Then minimizing the expected score implies = p. See (Lindley, 1982) for more details. This subjectivist interpretation is close to the everyday usage of the term “probability.” Yet appreciation of the subjectivist interpretation of probability is not wide spread in economics.
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  • Fall '10
  • Probability theory, probability density function, Cumulative distribution function, random quantities, D. A. Freedman

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Poirier-Handbook+Final - Exchangeability Representation...

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