A new method is introduced for characterizing and analyzing materials with random heterogeneous microstructure. The method begins with classifiers which process information from high-fidelity analyses of small-sized simulated microstructures. These classifiers are subsequently used in a multipass moving window to identify subregions of potentially critical microscale behavior such as strain concentrations. In the derivation of the method, it is shown how information theory-based concepts can be formulated in a Bayesian decision theory framework that addresses microstructural issues. Furthermore, it is shown how a sequence of classifiers can be constructed to refine the analysis of microstructure. While the method presented herein is general, a relatively simple example of a two-dimensional, two-phase composite is used to illustrate the analysis steps.