A Bernstein-Von Mises Theorem for discrete probability distributions

We investigate the asymptotic normality of the posterior distribution in the discrete setting, when model dimension increases with sample size. We consider a probability mass function theta(0) on N \ {0} and a sequence of truncation levels (k(n))(n) satisfying k(n)(3) <= n inf(i <= kn) theta(0)(i). Let (theta) over cap denote the maximum likelihood estimate of (theta(0)(i))(i <= kn) and let Delta(n)(theta(0)) denote the k(n)-dimensional vector which i-th coordinate is defined by root n((theta) over cap (n)(i)-theta(0)(i)) for 1 <= i <= k(n). We check that under mild conditions on theta(0) and on the sequence of prior probabilities on the k(n)-dimensional simplices, after centering and resealing, the variation distance between the posterior distribution recentered around (theta) over cap (n) and resealed by root n and the k(n)-dimensional Gaussian distribution N(Delta(n)(theta(0)), I(-1)(theta(0))) converges in probability to 0. This theorem can be used to prove the asymptotic normality of Bayesian estimators of Shannon and Renyi entropies. The proofs are based on concentration inequalities for centered and non-centered Chi-square (Pearson) statistics. The latter allow to establish posterior concentration rates with respect to Fisher distance rather than with respect to the Hellinger distance as it is commonplace in non-parametric Bayesian statistics.