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.
Highlights
Factor models describe a dependence structure among a high-dimensional correlated random vector in terms of a low-dimensional unobserved random vector called a latent factor or just factor
The main contribution of this paper is to propose a Bayesian factor model which is computationally tractable and at the same time achieves the posterior consistency of the factor dimensionality
If k0n ≤ Kn for some positive sequence {Kn}n∈N, it can be shown from our proof that the posterior consistency of the factor dimensionality is obtained with the hyperparameter αn p−n AsnKn for sufficiently large A > 0, but this yields the posterior contraction rate cn snk0nKn log pn/n of the covariance matrix, which is larger than the optimal one cn snk0n log pn/n when Kn → ∞
Summary
Factor models describe a dependence structure among a high-dimensional correlated random vector in terms of a low-dimensional unobserved random vector called a latent factor or just factor. Gao and Zhou (2015) studied a Bayesian sparse principal component analysis (PCA) model, which is equivalent to the factor model with the constraint that the columns of the loading matrix are orthogonal to each other They derived posterior contraction rates of the covariance matrix and the principal subspace estimation with respect to the spectral norm and proved the posterior consistency of the rank of the covariance matrix. The main contribution of this paper is to propose a Bayesian factor model which is computationally tractable and at the same time achieves the posterior consistency of the factor dimensionality For this purpose, we consider a spike and slab prior with.
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