A Large-Deviation Principle for Dirichlet Posteriors
Ayalvadi Ganesh, Neil O’Connell · Bernoulli · 2000
Let X k be a sequence of independent and identically distributed random variables taking values in a compact metric space\\Omega\\Gamma and consider the problem of estimating the law of X 1 in a Bayesian framework. A conjugate family of priors for non-parametric Bayesian inference is the Dirichlet process priors popularized by Ferguson. We prove that if the prior distribution is Dirichlet, then the sequence of posterior distributions satisfies a large deviation principle, and give an explicit expression for the rate function. As an application, we obtain an asymptotic formula for the predictive probability of ruin in the classical gambler's ruin problem. 1 Introduction Let X be a Hausdorff topological space with Borel oe-algebra B, and let ¯ n be a sequence of probability measures on (X ; B). A rate function is a nonnegative lower semicontinuous function on X . We say that the sequence ¯ n satisfies the large deviation principle (LDP) with rate function I, if for all B 2 B, \\Gamma inf...