Asymptotic Normality of Posterior Distributions

Suppose X 1,..., X n are independent observations from P θ , θ ∈ R. Suppose that P θ has density f θ (x) with respect to some measure v. The maximum likelihood estimate of θ (or the value of θ that maximizes the density of the posterior probability relative to the prior probability), maximizing Π i=1 n f θ (X i ) is denoted by \( {at heta _n}\) . As n → ∞, Fisher established that\( {at heta _n}\)is asymptotically normal with mean θ 0 and variance (nI(θ 0))−1, where θ 0 is the true value of θ, and I(θ 0) is Fisher’s information— \({ - ({d^2}/d{heta ^2}){P_{{heta _0}}}[og {f_heta }(X)] _{heta = {heta _0}}}\). The asymptotic normality requires a tedious list of regularity conditions, first promulgated by Wald.