Bernstein–von Mises theorem

In Bayesian inference, the Bernstein–von Mises theorem provides the basis for using Bayesian credible sets for confidence statements in parametric models. It states that under some conditions, a posterior distribution converges in total variation distance to a multivariate normal distribution centered at the maximum likelihood estimator θ ^ n {\displaystyle {\widehat {\theta }}_{n}} with covariance matrix given by n − 1 I ( θ 0 ) − 1 {\displaystyle n^{-1}{\mathcal {I}}(\theta _{0})^{-1}} , where θ 0 {\displaystyle \theta _{0}} is the true population parameter and I ( θ 0 ) {\displaystyle {\mathcal {I}}(\theta _{0})} is the Fisher information matrix at the true population parameter value: | | P ( θ | x 1 , … x n ) − N ( θ ^ n , n − 1 I ( θ 0 ) − 1 ) | | T V → P θ 0 = 0 {\displaystyle ||P(\theta |x_{1},\dots x_{n})-{\mathcal {N}}({\widehat {\theta }}_{n},n^{-1}{\mathcal {I}}(\theta _{0})^{-1})||_{\mathrm {TV} }\xrightarrow {P_{\theta _{0}}} =0} The Bernstein–von Mises theorem links Bayesian inference with frequentist inference.

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