Posterior distributions are computable from predictive distributions

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.