# Latent Mixture Modeling for Clustered Data

## TL;DR

This paper introduces a novel mixture modeling approach for clustered data, utilizing latent experts and Dirichlet-distributed mixing proportions, estimated via a Monte Carlo EM algorithm, with extensions for covariate dependence.

## Contribution

It develops a new mixture-of-experts model for clustered data with a Monte Carlo EM estimation method and covariate-dependent mixing proportions.

## Key findings

- The proposed model outperforms existing mixture models in simulations.
- It effectively captures cluster-specific distributions.
- Application to Japanese land price data demonstrates practical utility.

## Abstract

This article proposes a mixture modeling approach to estimating cluster-wise conditional distributions in clustered (grouped) data. We adapt the mixture-of-experts model to the latent distributions, and propose a model in which each cluster-wise density is represented as a mixture of latent experts with cluster-wise mixing proportions distributed as Dirichlet distribution. The model parameters are estimated by maximizing the marginal likelihood function using a newly developed Monte Carlo Expectation-Maximization algorithm. We also extend the model such that the distribution of cluster-wise mixing proportions depends on some cluster-level covariates. The finite sample performance of the proposed model is compared with some existing mixture modeling approaches as well as linear mixed model through the simulation studies. The proposed model is also illustrated with the posted land price data in Japan.