Bayesian inverse problems

Semiparametric Uncertainty Quantification via Isotonized Posterior for Deconvolutions

A novel nonparametric Bayesian approach for uncertainty quantification in the deconvolution model Z = X + Y, where the goal is to estimate the distribution of X from noisy observations. By placing a Dirichlet Process prior on the observed data and isotonizing posterior draws via the Greatest Convex Majorant, the Isotonic Inverse Posterior yields computationally fast credible sets with asymptotically correct frequentist coverage, without requiring estimation of any nuisance parameters.

Semiparametric Uncertainty Quantification via Isotonized Posterior for Deconvolutions

Semiparametric Bernstein-von Mises Phenomenon via Isotonized Posterior in Wicksell's problem

We propose a novel Bayesian approach for nonparametric estimation in Wicksell’s problem. We deviate from the classical Bayesian …

Francesco Gili, Aad van der Vaart, Geurt Jongbloed

Bernstein-von Mises phenomenon via Isotonized posterior in Wicksell's problem

We propose a new Bayesian nonparametric approach to Wicksell’s problem by placing a Dirichlet Process prior on the observables’ distribution. This simplifies computation via conjugacy and enables asymptotically efficient estimation through projection onto the space of square-integrable, increasing functions. The resulting Isotonized Inverse Posterior (IIP) satisfies a Bernstein–von Mises theorem with minimax variance, adapting to the true cdf’s Hölder continuity without requiring smoothness estimation.