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Unformatted text preview: Computable de Finetti measures Cameron E. Freer a , Daniel M. Roy b a Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA b Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA Abstract We prove a uniformly computable version of de Finetti’s theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify non-local state. Along the way, we prove that a distribution on the unit interval is computable if and only if its moments are uniformly computable. Key words: de Finetti’s theorem, exchangeability, computable probability theory, probabilistic programming languages, mutation 2010 MSC: 03D78, 60G09, 68Q10, 03F60, 68N18 1. Introduction The classical de Finetti theorem states that an exchangeable sequence of real random variables is a mixture of independent and identically distributed (i.i.d.) sequences of random variables. Moreover, there is an (almost surely unique) measure-valued random variable, called the directing random measure , condi- tioned on which the random sequence is i.i.d. The distribution of the directing random measure is called the de Finetti measure or the mixing measure . This paper examines the computable probability theory of exchangeable se- quences of real-valued random variables. We prove a uniformly computable ver- sion of de Finetti’s theorem, which implies that computable exchangeable se- quences of real random variables have computable de Finetti measures. The classical proofs do not readily effectivize; instead, we show how to directly com- pute the de Finetti measure (as characterized by the classical theorem) in terms of a computable representation of the distribution of the exchangeable sequence. Along the way, we prove that a distribution on [0 , 1] ω is computable if and only if its moments are uniformly computable, which may be of independent interest. A key step in the proof is to describe the de Finetti measure in terms of the moments of a set of random variables derived from the exchangeable sequence. October 27, 2009 When the directing random measure is (almost surely) continuous, we can show that these moments are computable, which suffices to complete the proof of the main theorem in this case. In the general case, we give a proof inspired by a randomized algorithm which succeeds with probability one. 1.1. Computable Probability Theory These results are formulated in the Turing-machine-based bit-model for com- putation over the reals (for a general survey, see Braverman and Cook [ 1 ]). This computational model has been explored both via the type-2 theory of effectiv- ity (TTE) framework for computable analysis, and via effective domain-theoretic representations of measures....
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.
- Spring '11