On The Gaussian Approximation To Bayesian Posterior Distributions

The present article derives the minimal number $N$ of observations needed to consider a Bayesian posterior distribution as Gaussian. Two examples are presented. Within one of them, a chi-squared distribution, the observable $x$ as well as the parameter $\xi$ are defined all over the real axis, in the other one, the binomial distribution, the observable $x$ is an entire number while the parameter $\xi$ is defined on a finite interval of the real axis. The required minimal $N$ is high in the first case and low for the binomial model. In both cases the precise definition of the measure $\mu$ on the scale of $\xi$ is crucial.