Stochastic a posteriori blockmodels: Construction and assessment

In 1983, Holland, Laskey, and Leinhardt, using the ideas of Holland and Leinhardt, and Fienberg and Wasserman, introduced the notion of a stochastic blockmodel. The mathematics for stochastic a priori blockmodels, in which exogenous actor attribute data are used to partition actors independently of any statistical analysis of the available relational data, have been refined by several researchers and the resulting models used by many. Attempts to simultaneously partition actors and to perform relational data analyses using statistical methods that yield stochastic a posteriori blockmodels are still quite rare. In this paper, we discuss some old suggestions for producing such posterior blockmodels, and comment on other new suggestions based on multiple comparisons of model parameters, log-linear models for ordinal categorical data, and correspondence analysis. We also review measures for goodness-of-fit of a blockmodel, and we describe a natural approach to this problem using likelihood-ratio statistics generated from a popular model for relational data.