Chapter 23 - Posterior probability maps

Abstract

This chapter 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 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 finessed 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 provide a natural, two-level hierarchy that enables an empirical Bayesian approach. In this section, we present the 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 effect in the same data. The approach adopted here is a natural extension of parametric empirical Bayes described in the previous chapter. The resulting model entails global shrinkage priors to inform the estimation of effects at each voxel or bin in the image. These global priors can be regarded as a special case of spatial priors in the more general spatiotemporal models for functional magnetic resonance imaging (fMRI) introduced in Chapter 25. To date, inference in neuroimaging has been restricted largely to classical inferences based upon statistical parametric maps (SPMs). The alternative approach is to use Bayesian or conditional inference based upon the posterior distribution of the activation given the data (Holmes and Ford, 1993) . This necessitates the specification of priors (i.e. the probability distribution of the activation). Bayesian inference requires the posterior distribution and therefore rests upon a posterior density analysis. A useful way to summarize this posterior density is to compute the probability that the activation exceeds some threshold. This computation represents a Bayesian inference about the effect, in relation to the specified threshold. We now describe an approach to computing posterior probability maps for activation effects or, more generally, treatment effects in imaging data sequences. This approach represents the simplest and most computationally expedient way of constructing PPMs.