# American Institute of Mathematical Sciences

October  2017, 11(5): 857-874. doi: 10.3934/ipi.2017040

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

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

Received  May 2016 Revised  November 2016 Published  July 2017

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.

Citation: T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems & Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040
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##### References:
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
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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