A New Bayesian Single Index Model with or without Covariates Missing at Random

For many biomedical, environmental, and economic studies, the single index model provides a practical dimension reaction as well as a good physical interpretation of the unknown nonlinear relationship between the response and its multiple predictors. However, widespread uses of existing Bayesian analysis for such models are lacking in practice due to some major impediments, including slow mixing of the Markov Chain Monte Carlo (MCMC), the inability to deal with missing covariates and a lack of theoretical justification of the rate of convergence of Bayesian estimates. We present a new Bayesian single index model with an associated MCMC algorithm that incorporates an efficient Metropolis-Hastings (MH) step for the conditional distribution of the index vector. Our method leads to a model with good interpretations and prediction, implementable Bayesian inference, fast convergence of the MCMC and a first-time extension to accommodate missing covariates. We also obtain, for the first time, the set of sufficient conditions for obtaining the optimal rate of posterior convergence of the overall regression function. We illustrate the practical advantages of our method and computational tool via reanalysis of an environmental study.