Bayesian inference for nonlinear models

In this part we continue to examine the inverse problem using Bayesian inference, but now look at the case where the simulation code (invoked in the likelihood) is nonlinear. There no longer exists an analytic expression for the posterior, and instead we need numerical methods to interrogate the posterior, and extract useful information from it.

First we observe that - although technically complete - our pdf expression for the posterior (from Bayes) is not particularly useful on its own, due to a combination of two factors. Firstly the high-cost of evaluating the pdf at a single parameter value (a cost equal to a model solve), and secondly the propensity of the posterior to be concentrated in a small region, whose location is a priori unknown. We decide that samples from the posterior would be a more useful representation of the posterior random-variable, and discuss method for obtaining samples: the inverse tranform (using the CDF - which we don't have in the Bayesian case); rejection sampling (highly inefficient); and finally a promising candidate, the Markov-chain Monte-Carlo method (McMC).

In Tutorial 4 we investigate a McMC method applied to a nonlinear inverse problem based on a Poisson equation with uncertain, spatially varying coefficient. This is closely related to the problem of subsurface flow (Darcy flow) with uncertain permeability.

Total time: 2:11:00