Section: Scientific Foundations
Bayesian model selection
The Bayesian approach to statistical problems is fundamentally probabilistic. A joint probability distribution is used to describe the relationships among all the unknowns and the data. Inference is then based on the posterior distribution i.e. the conditional probability distribution of the parameters given the observed data. Beyond the specification of the joint distribution, the Bayesian approach is automatic. Exploiting the internal consistency of the probability framework, the posterior distribution extracts the relevant information in the data and provides a complete and coherent summary of post-data uncertainty. Using the posterior to solve specific inference and decision problems is then straightforward, at least in principle. The select team is interested in applications of this Bayesian approach for model uncertainty problems where a large number of different models are under consideration. The joint distribution is obtained by introducing prior distributions on all the unknowns, here the parameters of each model and the models themselves, and then combining them with the distributions for the data. Conditioning on the data then induces a posterior distribution of model uncertainty that can be used for model selection and other inference and decision problems. This is the essential idea and it can be powerful. However, two major challenges confront its practical implementation: the specification of the prior distributions and the calculation of various posterior distributions.