Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 350 / 2 / a

The conditional mean estimator is the posterior meanprovided the first moment is finite. A maximum a posteriori estimate is any maximizer of the posterior density relative to Lebesgue measure:Existence of the posterior density alone does not guarantee a finite posterior mean or existence of a maximizing point.

For the stated multivariate normal distribution and independent standard Gaussian noise, the negative log of the posterior density, up to a constant, isIts precision matrix $A^{T}A+{\Sigma }^{-1}$ is a positive-definite matrix, so Gaussian conjugacy for a normal linear model givesThe two forms agree becauseThe covariance matrix is $(A^{T}A+{\Sigma }^{-1}{)}^{-1}$, and the maximum a posteriori estimate equals the posterior mean for this multivariate normal distribution.