Related papers: Bayesian inverse problems with Gaussian priors

Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian…

Bayesian inference paradigms are regarded as powerful tools for solution of inverse problems. However, when applied to inverse problems in physical sciences, Bayesian formulations suffer from a number of inconsistencies that are often…

Many inverse problems focus on recovering a quantity of interest that is a priori known to exhibit either discontinuous or smooth behavior. Within the Bayesian approach to inverse problems, such structural information can be encoded using…

We establish a general Bernstein--von Mises theorem for approximately linear semiparametric functionals of fractional posterior distributions based on nonparametric priors. This is illustrated in a number of nonparametric settings and for…

We study frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. The predictors are separated into many groups and the group structure is pre-determined. Two features of the…

We consider the inverse problem of recovering an unknown functional parameter $u$ in a separable Banach space, from a noisy observation $y$ of its image through a known possibly non-linear ill-posed map ${\mathcal G}$. The data $y$ is…

We consider the statistical inverse problem of recovering an unknown function $f$ from a linear measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the…

We present a survey of some of our recent results on Bayesian nonparametric inference for a multitude of stochastic processes. The common feature is that the prior distribution in the cases considered is on suitable sets of piecewise…

This paper studies quasi Bayesian estimation and uncertainty quantification for an unknown function that is identified by a nonparametric conditional moment restriction. We derive contraction rates for a class of Gaussian process priors.…

In a Bayesian inverse problem setting, the solution consists of a posterior measure obtained by combining prior belief, information about the forward operator, and noisy observational data. This measure is most often given in terms of a…

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when…

We continue the investigation of Bernstein-von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999-2028]. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined,…

In an indirect Gaussian sequence space model lower and upper bounds are derived for the concentration rate of the posterior distribution of the parameter of interest shrinking to the parameter value $\theta^\circ$ that generates the data.…

Neural networks are popular state-of-the-art models for many different tasks.They are often trained via back-propagation to find a value of the weights that correctly predicts the observed data. Although back-propagation has shown good…

This article shows that a large class of posterior measures that are absolutely continuous with respect to a Gaussian prior have strong maximum a posteriori estimators in the sense of Dashti et al. (2013). This result holds in any separable…

In Bayesian statistics, one's prior beliefs about underlying model parameters are revised with the information content of observed data from which, using Bayes' rule, a posterior belief is obtained. A non-trivial example taken from the…

We study full Bayesian procedures for high-dimensional linear regression under sparsity constraints. The prior is a mixture of point masses at zero and continuous distributions. Under compatibility conditions on the design matrix, the…

We consider the efficient inference of finite dimensional parameters arising in the context of inverse problems. Our setup is the observation of a transformation of an unknown infinite dimensional signal $f$ corrupted by statistical noise,…

The Bayesian approach to solving inverse problems relies on the choice of a prior. This critical ingredient allows the formulation of expert knowledge or physical constraints in a probabilistic fashion and plays an important role for the…

The present paper proposes a Bayesian framework for inverse problems that seamlessly integrates optimization and inversion to enable rapid surrogate modeling, accurate parameter inference, and rigorous uncertainty quantification. Bayesian…