11 Bayesian methods

Abstract

Following a review of the classical least squares approach to solving inverse problems, we introduce the Bayesian approach, which treats the model as a random variable with a probability distribution that we seek to estimate. A prior distribution for the model parameters is combined with the data to produce a posterior distribution for the model parameters. In special cases, the Bayesian approach produces solutions that are equivalent to the least squares, maximum likelihood, and Tikhonov regularization solutions. The maximum entropy method for selecting a prior distribution is discussed. Several examples of the Bayesian approach are presented. © 2005 Elsevier Inc. All rights reserved.