Exploring the reliability of Bayesian reconstructions Kenneth M. Hanson and Gregory S. Cunningham * Los Alamos National Laboratory, MS P940 Los Alamos, New Mexico 87545 USA kmh@lanl.gov cunning@lanl.gov Proc. SPIE 2434, pp. 416-4236 (1995) Medical Imaging 1996: Image Processing, M.H. Loew, ed. ABSTRACT The Bayesian approach allows one to combine measurement data with prior knowledge about models of reality to draw inferences about the validity of those models. The posterior probability quantifies the degree of certainty one has about those models. We propose a method to explore the reliability, or uncertainty, of specific features of a Bayesian solution. If one draws an analogy between the negative logarithm of the posterior and a physical potential, the gradient of this potential can be interpreted as a force that acts on the model. As model parameters are perturbed from their maximum a posteriori (MAP) values, the strength of the restoring force that drives them back to the MAP solution is directly related to the uncertainty in those parameter estimates. The correlations between the uncertainties of parameter estimates can be elucidated. Keywords: Bayesian analysis, uncertainty estimation, reliability, geometric models 1. INTRODUCTION Bayesian analysis provides the foundation for a rich environment in which to explore inferences about models from both data and prior knowledge through the posterior probability. In an attempt to reduce an analysis problem to a manageable size, the usual approach is to present a single instantiation of the object model as “the answer”, typically that which maximizes the posterior (the MAP solution). However, because of uncertainties in the measurements and/or because of a lack of sufficient data to define an unambiguous answer (in the absence of regularizing priors), 1 there may be no unique answer to many real analysis problems. Rather, innumerable solutions are allowable. Of course, some solutions are more probable than others. A prominent feature of the Bayesian approach is that it provides the probability of every possible solution, which, in a sense, ranks various solutions. The ability to ascertain the uncertainty or reliability of the answer remains a pressing issue, particularly when the number of parameters in the model is large. The traditional approach of specifying uncertainty, through the calculation of the covariance in the parameters, which includes the correlation between the uncertainties in any two parameters, does not provide much insight. Skilling et al. 2 suggested that one display a sequence of distinct solutions drawn from the posterior probability distribution. By viewing this random walk through the posterior distribution by means of a video loop, one gets a feeling for the uncertainty in a Bayesian solution. However, the calculational method used in that work was based on a Gaussian approximation of the posterior probability distribution in the neighborhood of the MAP solution. Later Skilling made progress in dealing with non-Gaussian distributions. 3 While the probabilistic display of Skilling et al. provides a general impression of the overall degree of variation possible in the solution, we desire a means to probe the uncertainty in the solution in a more directed manner. * Supported by the United States Department of Energy under contract number W-7405-ENG-36.