On the Gaussian Approximation to Bayesian Posterior Distributions

The present article derives the minimal number N of observations needed to approximate a Bayesian posterior distribution by Gaussian. derivation is based on an invariance requirement for likelihood . This defined Lie group that leaves src=image/13422860_01.gif> unchanged, when applied both observation(s) src=image/13422860_05.gif> and parameter src=image/13422860_02.gif> be estimated. It leads, in turn, class specific priors. In general, criterion Gaussian approximation found depend (i) Fisher information related src=image/13422860_01.gif>, (ii) lowest non-vanishing order Taylor expansion Kullback-Leibler distance between src=image/13422860_03.gif>, where src=image/13422860_04.gif> maximum-likelihood estimator src=image/13422860_02.gif>, given src=image/13422860_05.gif>. Two examples are presented, widespread various statistical analyses. first one, chi-squared distribution, all over real axis. other binomial observation binary number, while finite interval Analytic expressions required cases. necessary magnitude larger model (continuous src=image/13422860_05.gif>) than (binary src=image/13422860_05.gif>). difference traced back symmetry properties function We see considerable practical interest our results since normal basis parametric methods statistics widely used diverse areas research (education, medicine, physics, astronomy etc.). To have analytical whether applicable or not, appears relevant practitioners these fields.