Regularized-Likelihood Deconvolution with Posterior-Density Reconstruction
Abstract
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.
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BibTeX
@misc{marzio2026regularizedlikelihood,
title = {{Regularized-Likelihood Deconvolution with Posterior-Density Reconstruction}},
author = {Marco Di Marzio and Stefania Fensore and Chiara Passamonti and Serena Pulcini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.00240},
eprint = {2608.00240},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00240},
}
RIS
TY - UNPB
TI - Regularized-Likelihood Deconvolution with Posterior-Density Reconstruction
AU - Marco Di Marzio
AU - Stefania Fensore
AU - Chiara Passamonti
AU - Serena Pulcini
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2608.00240
UR - https://arxiv.org/abs/2608.00240
AB - 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.
ER -