Laplace's method in Bayesian inverse problems

• DOI: 10.48550/arxiv.1701.07989
• Publication Date: 2017-01-01
• Authors: Philipp Wacker
• Keywords: Gaussian measure; Hellinger distance; Laplace's method; Operator (biology); Solver

In a Bayesian inverse problem setting, the solution consists of posterior measure obtained by combining prior belief, information about forward operator, and noisy observational data. This is most often given in terms density with respect to reference high-dimensional (or infinite-dimensional) Banach space. Although Monte Carlo sampling methods provide way querying posterior, necessity evaluating operator many times (which will be costly PDE solver) prohibits this practice. For reason, practitioners choose suitable Gaussian approximation measure, procedure called Laplace's method. Once generated, lot easier sample from properties like moments are immediately acquired. paper derives attributed explicitly as second-order data-misfit functional, specifically infinite-dimensional setting. By use reverse Cauchy-Schwarz inequality we able bound Hellinger distance between its approximation.