Posterior Consistency of Factor Dimensionality in High-Dimensional Sparse Factor Models

Abstract

. Factor models aim to describe a dependence structure among high-dimensional random variables in terms of a low-dimensional unobserved random vector called a factor. One of the major practical issues of applying the factor model is to determine the factor dimensionality. In this paper, we propose a computationally feasible nonparametric prior distribution which achieves the posterior consistency of the factor dimensionality. We also derive the posterior contraction rate of the covariance matrix which is optimal when the factor dimensionality of the true covariance matrix is bounded. We conduct numerical studies that illustrate our theoretical results.