Factor analysis for a mixture of continuous and binary random variables

We propose a multivariate probability distribution that models a linear correlation between binary and continuous variables. The proposed distribution is a natural extension of the previously developed multivariate binary distribution based on Grassmann numbers. As an application of the proposed distribution, we develop a factor analysis for a mixture of continuous and binary variables. We also discuss improper solutions associated with maximum likelihood estimation of factor analysis. As a prescription to avoid improper solutions, we impose a constraint that the norms of each vector of a factor loading matrix are the same. We numerically validated the proposed factor analysis and norm constraint prescription by analyzing real datasets.