2016/09/23 by Bamdad Hosseini, Hosseini, Bamdad
Engineering · Mathematics · #35R30 #60B11 #62F99 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Numerical Analysis (math.NA) #Probability (math.PR) #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1609.07532
openalex publication_date 2016/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a new class of prior measures in connection to \ℓp\nregularization techniques when p \∈(0,1) which is based on the generalized\nGamma distribution. We show that the resulting prior measure is heavy-tailed,\nnon-convex and infinitely divisible. Motivated by this observation we discuss\nthe class of infinitely divisible prior measures and draw a connection between\ntheir tail behavior and the tail behavior of their L 'evy measures. Next, we\nuse the laws of pure jump L 'evy processes in order to define new classes of\nprior measures that are concentrated on the space of functions with bounded\nvariation. These priors serve as an alternative to the classic total variation\nprior and result in well-defined inverse problems. We then study the\nwell-posedness of Bayesian inverse problems in a general enough setting that\nencompasses the above mentioned classes of prior measures. We establish that\nwell-posedness relies on a balance between the growth of the log-likelihood\nfunction and the tail behavior of the prior and apply our results to special\ncases such as additive noise models and linear problems. Finally, we discuss\nsome of the practical aspects of Bayesian inverse problems such as their\nconsistent approximation and present three concrete examples of well-posed\nBayesian inverse problems with heavy-tailed or stochastic process prior\nmeasures.\n