A Variational EM Method for Mixed Membership Models for Rank Data: an Analysis of Public Policy Preferences

Rank data arise in a wide variety of contexts including survey results, consumer preferences, and political votes. In many of these cases, one might expect significant population heterogeneity. In this article, we examine survey data from Eurobarometer 34.1, in which respondents ranked policy priorities on addressing illegal drugs, alcoholism, and AIDS. Population heterogeneity arises when there are multiple sub-populations or profile groups underlying the individual rankings. A mixed membership models allow an individual's membership to reflect partial membership in more than one sub-population. Previous methods for fitting mixed membership models to rank data utilize an MCMC approach that assumes that equal sized sub-populations. We propose a variational EM approach for fitting mixed membership models which allow for fast approximate inference. In addition, our method explicitly estimates the sub-population sizes, allowing them to vary. We apply this method to the Eurobarometer 34.1 data, find interpretable sub-populations which generally agree with the "left vs right" classification of political ideologies.