2012/03/31 by Sergios Agapiou, Stig Larsson, Andrew M. Stuart
Computer Science · Decision Sciences · Mathematics · #Applied mathematics #Bayesian inference #Bayesian linear regression #Bayesian probability #Conjugate prior #Contraction (grammar) #Covariance #Covariance operator #Gaussian Processes and Bayesian Inference #Hilbert space #Inverse problem #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematical optimization #Mathematics #Posterior probability #Prior probability #Probabilistic and Robust Engineering Design #Statistics #math.ST #msc:35R30 #msc:45Q05 #msc:62C10 #msc:62G20 #stat.TH
paper · pdf · doi:10.1016/j.spa.2013.05.001
published as Stochastic Process. Appl. 123 (2013), 3828-3860
openalex publication_date 2013/05/13 · arxiv created 2013/08/02 · arxiv updated 2013/08/05 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.