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NONCOMPUTABLE CONDITIONAL DISTRIBUTIONS
"... Abstract. We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomaticall ..."
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Abstract. We study the computability of conditional probability, a fundamental notion in probability theory and Bayesian statistics. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In more general settings, conditional probability is defined axiomatically, and the search for more constructive definitions is the subject of a rich literature in probability theory and statistics. However, we show that in general one cannot compute conditional probabilities. Specifically, we construct a pair of computable random variables (X, Y) in the unit interval whose conditional distribution P[YX] encodes the halting problem. Nevertheless, probabilistic inference has proven remarkably successful in practice, even in infinitedimensional continuous settings. We prove several results giving general conditions under which conditional distributions are computable. In the discrete or dominated setting, under suitable computability hypotheses, conditional distributions are computable. Likewise, conditioning is a computable operation in the presence of certain additional structure, such as independent absolutely continuous noise.
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"... As we devise more complicated prior distributions, will inference algorithms keep up? We highlight a negative result in computable probability theory by Ackerman, Freer, and Roy (2010) that shows that there exist computable priors with noncomputable posteriors. In addition to providing a brief surve ..."
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As we devise more complicated prior distributions, will inference algorithms keep up? We highlight a negative result in computable probability theory by Ackerman, Freer, and Roy (2010) that shows that there exist computable priors with noncomputable posteriors. In addition to providing a brief survey of computable probability theory geared towards the A.I. and statistics community, we give a new result characterizing when conditioning is computable in the setting of exchangeable sequences, and provide a computational perspective on work by Orbanz (2010) on conjugate nonparametric models. In particular, using a computable extension of de Finetti’s theorem (Freer and Roy 2009), we describe how to transform a posterior predictive rule for generating an exchangeable sequence into an algorithm for computing the posterior distribution of the directing random measure. 1
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"... As we devise more complicated prior distributions, will inference algorithms keep up? We highlight a negative result in computable probability theory by Ackerman, Freer, and Roy (2010) that shows that there exist computable priors with noncomputable posteriors. In addition to providing a brief surve ..."
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As we devise more complicated prior distributions, will inference algorithms keep up? We highlight a negative result in computable probability theory by Ackerman, Freer, and Roy (2010) that shows that there exist computable priors with noncomputable posteriors. In addition to providing a brief survey of computable probability theory geared towards the A.I. and statistics community, we give a new result characterizing when conditioning is computable in the setting of exchangeable sequences, and provide a computational perspective on work by Orbanz (2010) on conjugate nonparametric models. In particular, using a computable extension of de Finetti’s theorem (Freer and Roy 2009), we describe how to transform a posterior predictive rule for generating an exchangeable sequence into an algorithm for computing the posterior distribution of the directing random measure. 1