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Well-posed Bayesian inverse problems with infinitely divisible and heavy-tailed prior measures. (English) Zbl 1390.35417

Summary: We present a new class of prior measures based on the generalized Gamma distribution that are closely related to \(\ell_p\)-regularization techniques when \(p\in(0,1)\). Furthermore, we use the laws of pure jump Lévy processes in order to define new classes of prior measures that are concentrated on the space of functions with bounded variation. These priors serve as an alternative to the classic total variation prior and result in well-defined inverse problems. Some of these prior measures are heavy-tailed, nonconvex, and infinitely divisible. Motivated by this observation we study the class of infinitely divisible prior measures and draw a connection between their tail behavior and the tail behavior of their Lévy measures. We then study the well-posedness of Bayesian inverse problems in a general enough setting that encompasses the above-mentioned classes of prior measures. We establish that well-posedness relies on a balance between the growth of the log-likelihood function and the tail behavior of the prior and apply our results to special cases such as additive noise models and linear problems. Finally, we discuss some of the practical aspects of Bayesian inverse problems such as their consistent approximation and present three concrete examples of well-posed Bayesian inverse problems with heavy-tailed or stochastic process prior measures.

MSC:

35R30 Inverse problems for PDEs
62F99 Parametric inference
60B11 Probability theory on linear topological spaces

Software:

PhaseLift
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References:

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