Bayesian inference and posterior probability maps

This paper describes the construction of posterior probability maps that enable conditional or Bayesian inferences about regionally-specific effects in neuroimaging. Posterior probability maps are images of the probability or confidence that an activation or effect exceeds some specified threshold, given the data. Posterior probability maps (PPMs) represent a complementary alternative to statistical parametric maps (SPMs) that are used to make classical inferences. However, a key problem in Bayesian inference is the specification of appropriate priors. This problem can be eschewed by using empirical Bayes in which prior variances are estimated from the data, under some simple assumptions about their form. Empirical Bayes requires a hierarchical observation model, in which higher levels can be regarded as providing prior constraints on lower levels. In neuroimaging observations of the same effect over voxels provides a natural, two-level hierarchy that enables an empirical Bayesian approach. We present a brief motivation and the operational details of a simple empirical Bayesian method for computing posterior probability maps. We then compare Bayesian and classical inference through the equivalent PPMs and SPMs testing for the same effects in the same data.