Discretization-invariant Bayesian inversion and Besov space priors

2009/01/27 by Matti Lassas. Eero Saksman, Saksman, Matti Lassas. Eero, Samuli Siltanen +1 · 9 citations
Computer Science · Engineering · Mathematics · #42C40 #60F17 #65C20 #FOS: Computer and information sciences #FOS: Mathematics #Image and Signal Denoising Methods #Methodology (stat.ME) #Numerical methods in inverse problems #Probability (math.PR) #Sparse and Compressive Sensing Techniques

paper · pdf · doi:10.48550/arxiv.0901.4220

openalex publication_date 2009/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Bayesian solution of an inverse problem for indirect measurement M = AU + E is considered, where U is a function on a domain of R^d. Here A is a smoothing linear operator and E is Gaussian white noise. The data is a realization m_k of the random variable M_k = P_kA U+P_k E, where P_k is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as U_n=T_nU, where T_n is a finite dimensional projection, leading to the computational measurement model M_kn=P_k A U_n + P_k E. Bayes formula gives then the posterior distribution π_kn(u_n | m_kn)∼π_n(u_n) exp(-1/2‖m_kn - P_kA u_n‖₂²) in R^d, and the mean U^CM_kn:=∫ u_n π_kn(u_n | m_k) du_n is considered as the reconstruction of U. We discuss a systematic way of choosing prior distributions \prior_n for all n≥ n₀>0 by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is \em discretization-invariant in the sense that \prior_n represent the same \em a priori information for all n and that the mean U^CM_kn converges to a limit estimate as k,n→∞. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with B¹₁₁ prior is related to penalizing the ℓ¹ norm of the wavelet coefficients of U.

Citations

• Foundations of Modern Probability

Cited by

• Hierarchical Besov-Laplace priors for spatially inhomogeneous binary classification
• Discrete to continuum limits in Bayesian inverse problems
• Bayesian inference with Besov-Laplace priors for spatially inhomogeneous binary classification surfaces
• Sampling theorems for inverse problems on Riemannian manifolds
• Consistency of variational inference for Besov priors in non-linear inverse problems
• Laplace's method in Bayesian inverse problems
• Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method
• Posterior contraction rates of computational methods for Bayesian data assimilation
• Semiparametric Bernstein-von Mises theorems for reversible diffusions

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