A Bernstein-Von Mises Theorem for discrete probability distributions

Abstract

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 $θ_0$ on $\mathbbm{N}\setminus \{0\}$ and a sequence of truncation levels $(k_n)_n$ satisfying $k_n^3\leq n\inf_{i\leq k_n}θ_0(i).$ Let $\hatθ$ denote the maximum likelihood estimate of $(θ_0(i))_{i\leq k_n}$ and let $Δ_n(θ_0)$ denote the $k_n$-dimensional vector which $i$-th coordinate is defined by \sqrt{n} (\hatθ_n(i)-θ_0(i)) for $1\leq i\leq k_n.$ We check that under mild conditions on $θ_0$ and on the sequence of prior probabilities on the $k_n$-dimensional simplices, after centering and rescaling, the variation distance between the posterior distribution recentered around $\hatθ_n$ and rescaled by $\sqrt{n}$ and the $k_n$-dimensional Gaussian distribution $\mathcal{N}(Δ_n(θ_0),I^{-1}(θ_0))$ converges in probability to $0.$ This theorem can be used to prove the asymptotic normality of Bayesian estimators of Shannon and Rényi 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.