Bayesian inference for multivariate probit model with latent envelope

The response envelope model proposed by Cook et al. (2010) is an efficient method to estimate the regression coefficient under the context of the multivariate linear regression model. It improves estimation efficiency by identifying material and immaterial parts of responses and removing the immaterial variation. The response envelope model has been investigated only for continuous response variables. In this paper, we propose the multivariate probit model with latent envelope, in short, the probit envelope model, as a response envelope model for multivariate binary response variables. The probit envelope model takes into account relations between Gaussian latent variables of the multivariate probit model by using the idea of the response envelope model. We address the identifiability of the probit envelope model by employing the essential identifiability concept and suggest a Bayesian method for the parameter estimation. We illustrate the probit envelope model via simulation studies and real-data analysis. The simulation studies show that the probit envelope model has the potential to gain efficiency in estimation compared to the multivariate probit model. The real data analysis shows that the probit envelope model is useful for multi-label classification.