Bayesian Variational Inference for Mixed Data Mixture Models

Heterogeneous, mixed type datasets including both continuous and categorical variables are ubiquitous, and enriches data analysis by allowing for more complex relationships and interactions to be modelled. Mixture models offer a flexible framework for capturing the underlying heterogeneity and relationships in mixed type datasets. Most current approaches for modelling mixed data either forgo uncertainty quantification and only conduct point estimation, and some use MCMC which incurs a very high computational cost that is not scalable to large datasets. This paper develops a coordinate ascent variational inference algorithm (CAVI) for mixture models on mixed (continuous and categorical) data, which circumvents the high computational cost of MCMC while retaining uncertainty quantification. We demonstrate our approach through simulation studies as well as an applied case study of the NHANES risk factor dataset. We provide theoretical justification for our method by establishing that the CAVI variational posterior mean converges locally to the true parameter value at a gap of $O(1/n)$ from the maximum likelihood estimator. Building on this result, we show that the CAVI variational posterior contracts around the true parameter at $O(n^{-1/2})$ rate.