 Solution

For a Laplace prior $\varphi (u)\propto e^{-\lambda |u|}$, maximizing the posterior is equivalent to minimizingso the posterior mode is the Lasso. For a Gaussian prior $\varphi (u)\propto e^{-\lambda u^2/2}$, the mode minimizesand is the ridge regression estimator $(X^{T}X+\lambda I{)}^{-1}{X}^{T}Y$. Gaussian conjugacy makes the posterior normal, so its mode and posterior mean coincide at this ridge estimate.