Identifiable factor analysis for mixed continuous and binary variables based on the Gaussian-Grassmann distribution

TL;DR

This paper introduces a novel factor analysis method for mixed continuous and binary data using the Gaussian-Grassmann distribution, enabling analytical marginalization and improved model identifiability.

Contribution

It develops an identifiable factor analysis framework for mixed data types based on the Gaussian-Grassmann distribution, with a new norm constraint to prevent improper solutions.

Findings

Analytical marginalization of latent variables is achieved.

The model's identifiability is proven under the norm constraint.

The proposed method outperforms quantification in correlation reproducibility.

Abstract

We develop a factor analysis for mixed continuous and binary observed variables. To this end, we utilized a recently developed multivariate probability distribution for mixed-type random variables, the Gaussian-Grassmann distribution. In the proposed factor analysis, marginalization over latent variables can be performed analytically, yielding an analytical expression for the distribution of the observed variables. This analytical tractability allows model parameters to be estimated using standard gradient-based optimization techniques. We also address improper solutions associated with maximum likelihood factor analysis. We propose a prescription to avoid improper solutions by imposing a constraint that row vectors of the factor loading matrix have the same norm for all features. Then, we prove that the proposed factor analysis is identifiable under the norm constraint. We demonstrate the validity of this norm constraint prescription and numerically verified the model's identifiability using both real and synthetic datasets. We also compare the proposed model with quantification method and found that the proposed model achieves better reproducibility of correlations than the quantification method.