Nonparametric Priors on Complete Separable Metric Spaces

A Bayesian model is nonparametric if its parameter space has infinite dimension; typical choices are spaces of discrete measures and Hilbert spaces. We consider the construction of nonparametric priors when the parameter takes values in a more general functional space. We (i) give a Prokhorov-type representation result for nonparametric Bayesian models; (ii) show how certain tractability properties of the nonparametric posterior can be guaranteed by construction; and (iii) provide an ergodic decomposition theorem to characterize conditional independence when de Finetti’s theorem is not applicable. Our work is motivated primarily by statistical problems where observations do not form exchangeable sequences, but rather aggregate into some other type of random discrete structure. We demonstrate applications to two such problems, permutation-valued and graphvalued observations, and relate our results to recent work in discrete analysis and ergodic theory.