Bayesian Single Index Model with Covariates Missing at Random

Bayesian single index model is a highly promising dimension reduction tool for an interpretable modeling of the non linear relationship between the response and its predictors. However, existing Bayesian tools in this area suffer from slow mixing of the Markov Chain Monte Carlo (MCMC) computational tool and also lack the ability to deal with missing covariates. To circumvent these practical problems, we present a new Bayesian single index model with MCMC algorithm using a mode-alignment based proposal density for the index vector for an efficient Metropolis Hastings (MH) algorithm to sample from full conditional distribution. Our method leads to an interpretable model and inference, the efficient evaluation of the likelihood, fast convergence of the MCMC, and a first time extension of inference to missing at random covariates. We also prove the posterior consistency of the overall regression function which has been done for the first time in Bayesian single index model literature.