High-dimensional Bernstein–von Mises theorem for the Diaconis–Ylvisaker prior

Abstract

We study the asymptotic normality of the posterior distribution of canonical parameter in the exponential family under the Diaconis–Ylvisaker prior which is a conjugate prior when the dimension of parameter space increases with the sample size. We prove under mild conditions on the true parameter value ${\theta }_0$ and hyperparameters of priors, the difference between the posterior distribution and a normal distribution centered at the maximum likelihood estimator, and variance equal to the inverse of the Fisher information matrix goes to 0 in the expected total variation distance. The proof assumes dimension of parameter space $d$ grows linearly with sample size $n$ only requiring $d=o\left(n\right)$. En route, we derive a concentration inequality of the quadratic form of the maximum likelihood estimator without any specific assumption such as sub-Gaussianity. A specific illustration is provided for the Multinomial-Dirichlet model with an extension to the density estimation and Normal mean estimation problems.