# Multivariate mixed membership modeling: Inferring domain-specific risk profiles

## TL;DR

This paper introduces a multivariate mixed membership model that improves interpretability and fit by accounting for domain-specific variable blocks and cross-domain correlations, demonstrated through a malaria risk study.

## Contribution

The paper proposes a novel multivariate mixed membership model that explicitly models domain-based variable blocks and correlations, enhancing interpretability and fit over standard models.

## Key findings

- Fewer profiles are needed for good data fit.
- The model captures cross-domain correlations effectively.
- Applied successfully to malaria risk data.

## Abstract

Characterizing the shared memberships of individuals in a classification scheme poses severe interpretability issues, even when using a moderate number of classes (say 4). Mixed membership models quantify this phenomenon, but they typically focus on goodness-of-fit more than on interpretable inference. To achieve a good numerical fit, these models may in fact require many extreme profiles, making the results difficult to interpret. We introduce a new class of multivariate mixed membership models that, when variables can be partitioned into subject-matter based domains, can provide a good fit to the data using fewer profiles than standard formulations. The proposed model explicitly accounts for the blocks of variables corresponding to the distinct domains along with a cross-domain correlation structure, which provides new information about shared membership of individuals in a complex classification scheme. We specify a multivariate logistic normal distribution for the membership vectors, which allows easy introduction of auxiliary information leveraging a latent multivariate logistic regression. A Bayesian approach to inference, relying on P\'olya gamma data augmentation, facilitates efficient posterior computation via Markov Chain Monte Carlo. We apply this methodology to a spatially explicit study of malaria risk over time on the Brazilian Amazon frontier.