Asymptotic Normality of Posterior Distributions
Thomas S. Ferguson · 2017
Bayes estimates provide another class of asymptotically efficient estimates. We assume that θ is chosen from Θ an open subset of ℝ k according to a prior density g ( θ ) with respect to Lebesgue measure, d θ , and that g ( θ ) is continuous and positive on Θ. The posterior density of θ , given a sample X 1 ,…, X n from f ( x | θ ), is g ( θ | x 1 , … , x n ) = ( ∏ 1 n f ( x j θ ) ) g ( θ ) ∫ Θ ( ∏ 1 n f ( x j θ ) ) g ( θ ) d θ = L n ( θ ) g ( θ ) ∫ Θ L n ( θ ) g ( θ ) d θ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136288/0151e1f8-e710-445f-85b6-a1b9718c8993/content/eq1012.tif"/>