= Laplace prior for sparse images
{c}
{title2=$\pi_0(u)\propto e^{-\lambda\|u\|_1}$}
Independent centered <Laplace distributions> for pixel intensities give the <prior distribution>
$$
\pi_0(u)=(\lambda/2)^d e^{-\lambda\sum_{j=1}^d|u_j|},\qquad\lambda>0.
$$
This prior favors small intensities through its sharp peak at zero, while its tails permit isolated appreciable pixels. Its negative log density is an <L1 norm> penalty, so with a <Gaussian likelihood> a <maximum a posteriori estimate> minimizes a sum of squared residuals plus that penalty. A draw from this continuous prior has no exactly zero coordinate with positive probability; sparsity of a maximizing estimate and concentration of prior mass near zero are different statements. The product prior also does not encode spatial contiguity. <Inverse transform sampling> produces each coordinate from an independent <uniform distribution> by $\lambda^{-1}\log(2U)$ for $U\leq1/2$ and $-\lambda^{-1}\log(2(1-U))$ otherwise, ignoring the probability-zero endpoints.
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