Well-posed Bayesian Inverse Problems with Infinitely-Divisible and Heavy-Tailed Prior Measures

We present a new class of prior measures in connection to $\ell_p$ regularization techniques when $p \in(0,1)$ which is based on the generalized Gamma distribution. We show that the resulting prior measure is heavy-tailed, non-convex and infinitely divisible. Motivated by this observation we discuss the class of infinitely divisible priors and draw a connection between their tail behavior and the tail behavior of their L\'evy measures. Next, we study the well-posedness of Bayesian inverse problems with heavy-tailed prior measures on Banach spaces. 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, linear problems and infinitely divisible prior measures. Finally, we study some practical aspects of Bayesian inverse problems such as their consistent approximation and present three concrete examples of well-posed Bayesian inverse problems with infinitely-divisible prior measures.