We discuss inference for graphical models as a multiple comparison problem. We argue that posterior inference under a suitable hierarchical model can adjust for the multiplicity problem that arises by deciding inclusion for each of many possible edges. We show that inference under that hierarchical model differs substantially from inference under a comparable nonhierarchical model. We discuss several stylized inference problems, including estimation of one graph, comparison of a pair of graphs, estimation of a pair of graphs and, finally, estimation for multiple graphs. Throughout the discussion we the graph to identify conditional independence structure. Conditional on the graph a sampling model is proposed for the observed data. Most of the discussion is general and remains valid for any sampling model.
The discussion is motivated by two case studies. The first application is to model single cell mass spectrometry data for inference about the joint distribution of a set of markers that are recorded for each cell. Another application is to model RPPA protein expression data.
