On the statistical analysis of smoothing by maximizing dirty Markov random field posterior distributions

We consider Bayesian nonpararnetric function estimation using a Markov random field prior based on the Laplace distribution. We describe efficient methods for finding the exact maximum a posteriori estimate, which handle constraints naturally and avoid the problems posed by nondifferentiability of the posterior distribution; the methods also make links to spline and wavelet smoothers and to a dual posterior distribution. Three automatic smoothing parameter selection procedures are described: empirical Bayes, two-fold cross-validation, and a universal rule for the Laplace prior. Monte Carlo Simulation with Gaussian and Poisson responses demonstrates that the flew estimator can give better estimates of nonsmooth functions than can a similar prior based on the Gaussian distribution or wavelet-based competitors. Applications are given to spectral density estimation and to Poisson image denoising.