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Title: Normal approximation for the posterior in exponential families Authors:  Adrian Fischer - Université libre de Bruxelles (Belgium) [presenting]
Gesine Reinert - Oxford University (United Kingdom)
Robert Gaunt - The University of Manchester (United Kingdom)
Yvik Swan - Universite libre de Bruxelles (Belgium)
Abstract: Under suitable regularity conditions, the asymptotic normality of the posterior distribution is a fundamental result in Bayesian statistics, often referred to as the Bernstein-von Mises Theorem. In particular, it follows that the contribution of the prior distribution to the posterior distribution becomes negligible for large sample sizes $n$. We use Stein's method to obtain explicit bounds to quantify the multivariate normal approximation of the posterior distribution in multi-parameter exponential family models. We provide bounds of order $n^{-1/2}$ in the total variation (and thus Kolmogorov) and Wasserstein distances. Moreover, we apply our general bounds to several examples from exponential families, including Poisson likelihood with gamma prior, multinomial likelihood with Dirichlet prior, and the normal distribution with unknown mean and variance with normal-inverse gamma prior. These bounds are of the expected order $n^{-1/2}$ and have an explicit dependence on the parameters of the prior distribution and sufficient statistics of the data from the sample, and thus provide insight into how these factors affect the quality of the normal approximation. The performance of the bounds is also assessed with simulations.