Gaussian posterior representation of ridge regression ($\lambda ={\sigma }^2/{\tau }^2,V={\sigma }^2(X^{T}X+\lambda I{)}^{-1}$)

For likelihood $Y\mid \beta \sim N(X\beta ,{\sigma }^{2}I)$ and independent prior $\beta \sim N(0,{\tau }^{2}I)$, the posterior distribution is Gaussian with covariance $V$ as displayed and mean $(X^{T}X+\lambda I{)}^{-1}{X}^{T}Y$. Its mean and mode therefore coincide with the ridge regression coefficient estimate. A Gaussian prior alone does not imply this conclusion without the stated likelihood.

// Target: foundations-of-mathematics.bigb