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.

31 pages, minor correction to the proof of Proposition 3.5

References in corpus (8)

• Rates of contraction of posterior distributions based on Gaussian process priors
• Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
• Reproducing kernel Hilbert spaces of Gaussian priors
• Posterior convergence rates of Dirichlet mixtures at smooth densities
• Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems
• Global uniform risk bounds for wavelet deconvolution estimators
• Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$
• Bayesian optimal adaptive estimation using a sieve prior

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