Bayesian inverse problems with non-conjugate priors

By Kolyan Ray. 2012

89 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f81a2350b742116e6a478ecb5747cd331e3880aa

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Citations in the corpus

On the Frequentist Properties of Bayesian Nonparametric Methods (2016, background): “Ray, 2013.”
Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, methodology): “As noted by [23] for linear inverse problems Y = Af + ǫ, A : L(2) → L(2) linear, with unknown parameter f ∈ L2 and Gaussian white noise ǫ, a key point is to take advantage of the fact that the usual information distance ‖f‖L2 (when A = Id) is transformed to ‖Af‖L2 , which typically corresponds to a ”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “ing a hyperprior or an empirical Bayes method. In the case of Gaussian white noise, a number of Bayesian procedures have been shown to be rate adaptive over common smoothness classes (see for example [27,29,42]). Most such frequentist analyses restrict attention to obtaining contraction rates and do”
Analysis of the Gibbs Sampler for Hierarchical Inverse Problems (2013, background): “Finally, it gives rise to Gaussian posterior distributions which are well studied in the sense of posterior consistency; see [25, 3, 36].”
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, result): “In this regard our results are related to recent investigations of Bayesian inverse problems [29, 39, 4, 17, 28, 35], Bernstein-von Mises theorems [40, 32, 31, 33] and diffusion models [50, 38, 54, 34, 49, 26, 1].”
MAP estimators and their consistency in Bayesian nonparametric inverse problems (2013, background): “The papers [1, 23] are devoted to obtaining rates of convergence for linear inverse problems with conjugate Gaussian priors, whilst the papers [4, 29] study nonconjugate priors for linear inverse problems.”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “Regarding statistical (frequentist) properties of posterior measures, the case of linear G is again fairly well understood due to the explicit Gaussian structure of the posterior distribution, we refer here only to [35, 52, 2, 34, 39, 22, 26] and references therein.”
Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator (2015, methodology): “In the hierarchical setup with a prior on k, type (T1) prior has been studied by [1, 25] for generic models, by [22] for density estimation, by [2] for Gaussian white noise model and by [20] for inverse problems.”