Dealing with reflection invariance in Bayesian factor analysis

摘要

A well-known inherent ambiguity in factor models is that factors and factor loadings can only be identified up to an orthogonal rotation. This paper is concerned with a special case of rotations – reflections – that correspond to sign changes of the columns in the matrix of loadings, and the associated implications for Bayesian factor analysis. Using a classic data set as an example, we first demonstrate that naive implementation of rotational constraints can be problematic for Bayesian inference. Specifically, constraining some loadings to be one or positive to achieve identifiability may result in nontrivial multimodality. We present a simple approach for dealing with reflection invariance in Bayesian factor analysis. We recommend fitting Bayesian factor analysis models without rotational constraints on the loadings – allowing Markov chain Monte Carlo algorithms to explore the full posterior distribution – and then using a relabeling algorithm to pick a factor solution that corresponds to one mode. We demonstrate our approach on the case of a bifactor model, however, the relabeling algorithm is straightforward to transfer to other settings, and has already been applied in the literature for Bayesian estimation of other latent variable models.