Tree-Embedded Bayesian Factor Models for Multidimensional Categorical Distributions

TL;DR

This paper introduces a novel Bayesian latent factor model that uses a tree-based transformation to embed multidimensional categorical distributions into Euclidean space, enabling more flexible analysis of grouped data without strict parametric assumptions.

Contribution

The paper proposes a new nonparametric Bayesian factor model utilizing a tree-embedded transformation for better modeling of complex distributions in grouped data.

Findings

Outperforms standard Dirichlet mixture models in experiments

Effectively captures spatial dependence with SAR priors

Provides a flexible, nonparametric approach for distribution analysis

Abstract

Analyzing data collected from multiple sources to estimate common and heterogeneous structures through a hierarchical model is a central task in Bayesian inference, and to this end, Bayesian factor models are one of the most widely used tools for this purpose. In this paper, we propose a new Bayesian latent factor model for distributions, providing a parsimonious model for describing many observed distributions through lower-dimensional structures. Many applications are found in the social science in the form of grouped data, for example, distributions of age composition and income observed across locations. In these contexts, standard mixture models can be inefficient because the distributions do not necessarily exhibit clear clustering structures. To overcome the difficulty, we introduce a tree-based transformation that embeds distributions into a Euclidean space and construct a Bayesian latent factor model in the transformed space. Once transformed into Euclidean vectors, the Bayesian hierarchical model can be extended in a straightforward manner. As an illustration, we incorporate spatial dependence by introducing a prior based on a simultaneous autoregressive (SAR) model. The proposed model is "nonparametric" in the sense that it does not impose parametric assumptions on the form of the observed distributions. Through numerical experiments using real population data, we demonstrate that the proposed model outperforms the standard Dirichlet mixture model as well as models built on parametric assumptions.