2018/11/29 by Sergios Agapiou, Agapiou, Sergios, Masoumeh Dashti +3
Computer Science · Mathematics · #60G50 #62G05 #62G20 #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Markov Chains and Monte Carlo Methods #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1811.12244
openalex publication_date 2018/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a family of infinite dimensional product measures with tails\nbetween Gaussian and exponential, which we call p-exponential measures. We\nstudy their measure-theoretic properties and in particular their concentration.\nOur findings are used to develop a general contraction theory of posterior\ndistributions on nonparametric models with p-exponential priors in separable\nBanach parameter spaces. Our approach builds on the general contraction theory\nfor Gaussian process priors in van der Vaart and van Zanten 2008, namely we use\nprior concentration to verify prior mass and entropy conditions sufficient for\nposterior contraction. However, the specific concentration properties of\np-exponential priors lead to a more complex entropy bound which can influence\nnegatively the obtained rate of contraction, depending on the topology of the\nparameter space. Subject to the more complex entropy bound, we show that the\nrate of contraction depends on the position of the true parameter relative to a\ncertain Banach space associated to p-exponential measures and on the small\nball probabilities of these measures. For example, we apply our theory in the\nwhite noise model under Besov regularity of the truth and obtain minimax rates\nof contraction using (rescaled) \α-regular p-exponential priors. In\nparticular, our results suggest that when interested in spatially inhomogeneous\nunknown functions, in terms of posterior contraction, it is preferable to use\nLaplace rather than Gaussian priors.\n