Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors

T. J. Sullivan

doi:10.3934/ipi.2017040

Article Contents

2017, Volume 11, Issue 5: 857-874. Doi: 10.3934/ipi.2017040

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Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors

T. J. Sullivan

Free University of Berlin and Zuse Institute Berlin, Takustraße 7,14195 Berlin, Germany

Received: April 30, 2016

Revised: October 31, 2016

Published: September 2017

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Abstract

This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559,2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loéve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.

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Mathematics Subject Classification: Primary: 65J22; Secondary: 35R30, 60E07, 62F15, 62G35, 60B11, 28C20.

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Figure 1. Uniform angular measure on a circle projects radially to give Cauchy measure with width parameter $\gamma$ on any line at distance $\gamma$ from the centre of the circle

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Figure 2. Cauchy and Gaussian wavelet expansions in the linear spline orthonormal basis of $L^{2}([0, 1], \mathrm{d} x)$. Each horizontal stripe shows a random function $u(x) = \sum_{j = 0}^{J} \sum_{k = 0}^{2^{j} - 1} u_{j, k} 2^{j / 2} \psi(2^{j} x - k)$, where each $u_{j , k} = (j + 1)^{-2} 2^{-j}$ times a standard Cauchy or normal draw, and $\psi$ denotes the mother wavelet. The plots show $20$ i.i.d. samples with $J = 10$. Theorem 3.4 ensures a.s. convergence in $L^{2}([0, 1])$ as $J \to \infty$. To enable easy comparisons between plots, the ensemble has been translated and linearly scaled to take values $u(x) \in [0, 1]$, and the same random seed is used in each case

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