Chapter 3: Bayesian Inverse Problems

Learning Objectives

• Formulate an inverse problem as a Bayesian inference problem with prior, likelihood, and posterior
• Derive MAP and MMSE estimates from the posterior and explain how they relate to variational regularization
• Choose and justify sparsity-promoting priors (Laplace, spike-and-slab, horseshoe) for RF imaging scenes
• Implement sparse Bayesian learning (SBL) and automatic relevance determination (ARD) via the EM algorithm
• State the Cameron-Martin theorem and Stuart's well-posedness result for Gaussian measures on function spaces
• Compute posterior credible intervals and variance maps, and apply scalable uncertainty quantification methods
• Run MCMC samplers (MH, pCN, Gibbs, HMC) for posterior inference and assess their dimension scaling