= Solution
For a Laplace prior $\phi(u)\propto e^{-\lambda|u|}$, maximizing the posterior is equivalent to minimizing
$$
\frac12\|Y-X\beta\|_2^2+\lambda\|\beta\|_1,
$$
so the posterior mode is the <Lasso>. For a Gaussian prior $\phi(u)\propto e^{-\lambda u^2/2}$, the mode minimizes
$$
\frac12\|Y-X\beta\|_2^2+\frac\lambda2\|\beta\|_2^2
$$
and is the <ridge regression> estimator $(X^TX+\lambda I)^{-1}X^TY$. Gaussian conjugacy makes the posterior normal, so its mode and <posterior mean> coincide at this ridge estimate.
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