Consistency of Bayesian inference with Gaussian process priors in an\n elliptic inverse problem

2019/10/16 by Matteo Giordano, Giordano, Matteo, Richard Nickl +1 · 3 citations
Mathematics · Computer Science · #Statistical Methods and Inference #Gaussian Processes and Bayesian Inference #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.1910.07343

Abstract

For \O a bounded domain in \ℝ^d and a given smooth\nfunction g:\O\→\ℝ, we consider the statistical nonlinear\ninverse problem of recovering the conductivity f>0 in the divergence form\nequation \n
nabla
cdot(f
nabla u)=g

textrmon

mathcalO,
quad\n u=0

textrmon

partial
mathcalO, from N discrete noisy point\nevaluations of the solution u=u_f on mathcal O. We study the statistical\nperformance of Bayesian nonparametric procedures based on a flexible class of\nGaussian (or hierarchical Gaussian) process priors, whose implementation is\nfeasible by MCMC methods. We show that, as the number N of measurements\nincreases, the resulting posterior distributions concentrate around the true\nparameter generating the data, and derive a convergence rate N^-\λ,\n\λ>0, for the reconstruction error of the associated posterior means, in\nL²(\O)-distance.\n

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