Issues with Bayesian Analysis of Inverse Problems

A computer/math model of a process often contains numerous unknown parameters. To utilize the model, these parameters either must be determined from expert knowledge or estimated from data. In the mathematical community, the most common methods of estimating such parameters are through regularized best fit of various forms, and these generally work fine in the presence of plentiful data. However, when the data is limited, as is typically the case, parameter estimates obtained in this way can be very uncertain and use of the model without recognizing this uncertainty is quite problematic. Bayesian methods of solving such inverse problems have been growing in popularity, because of their ability to account for this uncertainty. A number of issues and problems that arise in implementing Bayesian methods will be discussed.