Well-Posed Bayesian Inverse Problems: Priors with Exponential Tails

Hosseini, Bamdad; Nigam, Nilima

doi:10.1137/16m1076824

Cited by 28 publications

(34 citation statements)

References 33 publications

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“…Stability of the posterior with respect to the observed data y and the log-likelihood Φ was established for Gaussian priors by Stuart (2010) and for more general priors by many later contributions (Dashti et al, 2012;Hosseini, 2017;Hosseini and Nigam, 2017;Sullivan, 2017). (We note in passing that the stability of BIPs with respect to perturbation of the prior is possible but much harder to establish, particularly when the data y are highly informative and the normalisation constant Z(y) is close to zero; see e.g.…”

Section: Bayesian Inverse Problemsmentioning

confidence: 99%

Random Forward Models and Log-Likelihoods in Bayesian Inverse Problems

Lie¹,

Sullivan²,

Teckentrup³

2018

SIAM/ASA J. Uncertainty Quantification

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We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations.

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Section: Bayesian Inverse Problemsmentioning

confidence: 99%

Random Forward Models and Log-Likelihoods in Bayesian Inverse Problems

Lie¹,

Sullivan²,

Teckentrup³

2018

SIAM/ASA J. Uncertainty Quantification

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“…satisfies (15). Unfortunately the reverse kernelQ * β in this case does not always have a closed form and must be identified on a case by case basis.…”

Section: The Sarsd Algorithmmentioning

confidence: 99%

“…Here G : X → R M is a deterministic forward map and Σ Σ Σ ∈ R M ×M is a positive-definite symmetric matrix. The additive Gaussian noise model above is widely used in practice [8,20,34] and it is the primary model in this article (see [15,20,34] for examples with other noise models). Using (2) we can readily identify µ u (y), the conditional probability measure of the data y given u, with Lebesgue density Here Λ denotes the Lebesgue measure and we used the familiar notation · Σ Σ Σ := Σ Σ Σ −1/2 · 2 .…”

Section: Introductionmentioning

confidence: 99%

Two Metropolis--Hastings Algorithms for Posterior Measures with Non-Gaussian Priors in Infinite Dimensions

Hosseini¹

2019

SIAM/ASA J. Uncertainty Quantification

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We introduce two classes of Metropolis-Hastings algorithms for sampling target measures that are absolutely continuous with respect to non-Gaussian prior measures on infinite-dimensional Hilbert spaces. In particular, we focus on certain classes of prior measures for which prior-reversible proposal kernels of the autoregressive type can be designed. We then use these proposal kernels to design algorithms that satisfy detailed balance with respect to the target measures. Afterwards, we introduce a new class of prior measures, called the Bessel-K priors, as a generalization of the gamma distribution to measures in infinite dimensions. The Bessel-K priors interpolate between well-known priors such as the gamma distribution and Besov priors and can model sparse or compressible parameters. We present concrete instances of our algorithms for the Bessel-K priors in the context of numerical examples in density estimation, finite-dimensional denoising and deconvolution on the circle.

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“…We now show that by Theorem 10 we obtain well-posedness w.r.t. the Wasserstein distance under the same basic assumption on Φ or , respectively, stated in [7,38] as well as slightly modified in [21,22,40] for establishing well-posedness w.r.t. the Hellinger distance.…”

Section: Remark 13 (Proofs Via Couplingsmentioning

confidence: 99%

On the local Lipschitz stability of Bayesian inverse problems

Sprungk

2020

Inverse Problems

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In this note we consider the robustness of posterior measures occuring in Bayesian inference w.r.t. perturbations of the prior measure and the log-likelihood function. This extends the well-posedness analysis of Bayesian inverse problems. In particular, we prove a general local Lipschitz continuous dependence of the posterior on the prior and the log-likelihood w.r.t. various common distances of probability measures. These include the Hellinger and Wasserstein distance and the Kullback-Leibler divergence. We only assume the boundedness of the likelihoods and measure their perturbations in an L p -norm w.r.t. the prior. The obtained robustness yields under mild assumptions the well-posedness of Bayesian inverse problems, in particular, a well-posedness w.r.t. the Wasserstein distance which is missing in the existing literature. Moreover, our results indicate an increasing sensitivity of Bayesian inference as the posterior becomes more concentrated, e.g., due to more or more accurate data. This confirms and extends previous observations made in the sensitivity analysis of Bayesian inference.

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Well-Posed Bayesian Inverse Problems: Priors with Exponential Tails

Cited by 28 publications

References 33 publications

Random Forward Models and Log-Likelihoods in Bayesian Inverse Problems

Random Forward Models and Log-Likelihoods in Bayesian Inverse Problems

Two Metropolis--Hastings Algorithms for Posterior Measures with Non-Gaussian Priors in Infinite Dimensions

On the local Lipschitz stability of Bayesian inverse problems

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