Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

Jia, Junxiong; Peng, Jigen; Gao, Jinghuai

doi:10.3934/ipi.2020061

Cited by 5 publications

(3 citation statements)

References 32 publications

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“…If G is a linear operator (e.g., examples in [23]), we can verify condition (3.38) easily. For specific nonlinear problems, we should employ the regularity properties of the forward equation, which is beyond the scope of the present work.…”

Section: Along the Research Line Of Newton-type Optimization Algorith...mentioning

confidence: 99%

Stein variational gradient descent on infinite-dimensional space and applications to statistical inverse problems

Jia¹,

Li²,

Meng³

2021

Preprint

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For solving Bayesian inverse problems governed by large-scale forward problems, we present an infinite-dimensional version of the Stein variational gradient descent (iSVGD) method, which has the ability to generate approximate samples from the posteriors efficiently. Specifically, we introduce the concept of the operator-valued kernel and the corresponding function-valued reproducing kernel Hilbert space (RKHS). Through the properties of RKHS, we give an explicit meaning of the infinite-dimensional objects (e.g., the Stein operator) and prove that the infinite-dimensional objects are indeed the limit of finite-dimensional items. Furthermore, by generalizing the change of variables formula, we construct iSVGD with preconditioning operators, yielding more efficient iSVGD. During these generalizations, we introduce a regularity parameter s ∈ [0, 1]. Our analysis shows that the intuitive trivial version (i.e., by directly taking finite-dimensional objects as infinite-dimensional items) of iSVGD with preconditioning operators (s = 0) will yield inaccurate estimates, and the parameter s should be chosen larger than 0 and smaller than 0.5. Finally, the proposed algorithms are applied to an inverse problem governed by the Helmholtz equation. Numerical results confirm the correctness of our theoretical findings and demonstrate the potential usefulness of the proposed approach in the posterior sampling of large-scale nonlinear statistical inverse problems.

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Section: Along the Research Line Of Newton-type Optimization Algorith...mentioning

confidence: 99%

Stein variational gradient descent on infinite-dimensional space and applications to statistical inverse problems

Jia¹,

Li²,

Meng³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Analysis of more general settings requires additional effort: NPIV is an inverse problem, and we anticipate the subtleties of model selection in nonparametric inverse problems. For example, analyses are usually restricted to the selection of γ [42,48,49], and the γ > 1 case requires additional assumptions [48]. 8 In the IV setting, it should also be noted that valid model comparison requires a good approximation to E| H , since otherwise the quasi-likelihood becomes less meaningful at any finite sample size.…”

Section: S2mentioning

confidence: 99%

Fast Instrument Learning with Faster Rates

Wang¹,

Zhou²,

Zhu³

2022

Preprint

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We investigate nonlinear instrumental variable (IV) regression given highdimensional instruments. We propose a simple algorithm which combines kernelized IV methods and an arbitrary, adaptive regression algorithm, accessed as a black box. Our algorithm enjoys faster-rate convergence and adapts to the dimensionality of informative latent features, while avoiding an expensive minimax optimization procedure, which has been necessary to establish similar guarantees. It further brings the benefit of flexible machine learning models to quasi-Bayesian uncertainty quantification, likelihood-based model selection, and model averaging. Simulation studies demonstrate the competitive performance of our method.Preprint. Under review.

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“…On the other hand, [33] extends the work of [3] by replacing the commutativity with a link condition. Jia et al in [23] construct an adaptive Bayes procedure with optimal contraction rates and without diagonal assumption.…”

Section: Introductionmentioning

confidence: 99%

A posterior contraction for Bayesian inverse problems in Banach spaces

Chen,

Li,

Zhang

2024

Inverse Problems

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This paper features a study of statistical inference for linear inverse problems with Gaussian noise and priors in structured Banach spaces. Employing the tools of sectorial operators and Gaussian measures on Banach spaces, we overcome the theoretical difficulty of lacking the bias-variance decomposition in Banach spaces, characterize the posterior distribution of solution though its Radon-Nikodym derivative, and derive the optimal convergence rates of the corresponding square posterior contraction and the mean integrated square error.Our theoretical findings are applied to two scenarios, specifically a Volterra integral equation and an inverse source problem governed by an elliptic partial differential equation. Our investigation demonstrates the superiority of our approach over classical results. Notably, our method achieves same order of convergence rates for solutions with reduced smoothness even in a Hilbert setting.

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Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

Cited by 5 publications

References 32 publications

Stein variational gradient descent on infinite-dimensional space and applications to statistical inverse problems

Stein variational gradient descent on infinite-dimensional space and applications to statistical inverse problems

Fast Instrument Learning with Faster Rates

A posterior contraction for Bayesian inverse problems in Banach spaces

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