Regularized-Likelihood Deconvolution with Posterior-Density Reconstruction
Density deconvolution is an ill-posed inverse problem because recovering the latent density amplifies high-frequency variation in the observed data. We propose a two-stage likelihood procedure. The first estimator maximizes the convolution likelihood over a regularized Gaussian-mixture sieve; the second treats this fit as an empirical prior and averages the fitted conditional latent densities. Under fixed-model misspecification, the estimators generally have different population targets. For bounded local multiplicative perturbations of the latent density, the population posterior map is locally contractive in chi-square divergence under identifiable convolution. Under correct finite-dimensional specification, however, reconstruction adds a nonnegative first-order variance component. For growing sieves, we separate observed-domain estimation, inverse stability, and posterior empirical variation. We derive explicit rates under Gaussian error and recover the classical ordinary-smooth exponent, up to logarithmic factors, with a constructive verification for Laplace error. Simulations and a Framingham blood-pressure application provide empirical performance evidences.