Bayesian inverse problems with non-conjugate priors

Abstract

We investigate the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform.Comment: 31 pages, minor correction to the proof of Proposition 3.

Bayes procedures for adaptive inference in inverse problems for the white noise model

“…The few papers in this area include [1,14,22,23,30]. Other papers addressing frequentist properties of Bayes procedures for different, but related inverse problems include [21] and [15].…”

Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

“…The subject has developing mathematical foundations and attendant stability and approximation theories [16,28,29,44,39]. Furthermore, the subject of Bayesian posterior consistency is being systematically developed [4,6,35,37,27,26,41,19,20]. Furthermore, the paper [25] was the first to establish consistency in the context of hyperparameter learning, as we do here, and in doing so demonstrates that Bayesian methods have comparable capabilities to frequentist methods, regarding adaptation to smoothness, whilst also quantifying uncertainty.…”

Bayesian inverse problems with unknown operators

“…If K ϑ is compact and admits an orthonormal basis of eigenfunction (e k ) k 1 being independent of ϑ, then this is assumption is trivially satisfied for (ϕ k ) = (e k ) and m = 0. On the other hand this assumption allows for more flexibility for the considered approximation spaces and can be compared to Condition 1 by Ray [30]. As a typical example, the possibly ϑ depended eigenfunctions (e ϑ,k ) of K ϑ may be the trigonometric basis of L 2 while V j are generated by bandlimited wavelets.…”

“…Both possibilities are only rarely studied for inverse problems. Using known operators, the few articles on this topic include Knapik et al [21] considering both constructions with a Gaussian prior on f and Ray [30] who has considered a sieve prior which could be interpreted as hierarchical Bayes procedure. We will follow the second path and choose J empirically using Lepski's method [24] which yields an easy to implement procedure (note that [21] used a maximum likelihood approach to estimate s).…”