Are Bayes Rules Consistent in Information

Bayes’ rule provides a method for constructing estimators of probability density functions in both parametric and nonparametric cases. Let X 1, X 2, ..., X n be a random sample from an unknown probability measure P 0 with density function p 0(x) with respect to a dominating measure λ(dx). Let µ be a prior probability measure on the space of all probability measures P which have densities p(x) = dP/dλ. Then the mean of the posterior yields the following estimator of the density function $${{\hat{p}}_{n}}(x) = \hat{p}(x;{{X}_{1}},{{X}_{2}}, \ldots ,{{X}_{n}}) = \frac{{\smallint p(x)(\prod _{{i = 1}}^{n}p({{X}_{i}}))d\mu }}{{\smallint (\prod _{{i = 1}}^{n}p({{X}_{i}}))d\mu }}.$$