= Gaussian posterior representation of ridge regression
{c}
{title2=$\lambda=\sigma^2/\tau^2,\quad V=\sigma^2(X^TX+\lambda I)^{-1}$}
For likelihood $Y\mid\beta\sim N(X\beta,\sigma^2I)$ and independent prior $\beta\sim N(0,\tau^2I)$, the <posterior distribution> is Gaussian with covariance $V$ as displayed and mean $(X^TX+\lambda I)^{-1}X^TY$. 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
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