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Abstract:
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Single index model is a popular tool in a wide variety of applications ranging from biomedical research to machine learning. From a frequentist viewpoint, there is a substantial amount of literature devoted to proposing strategies for estimating and inferring parameters of the model, with corresponding theoretical justification regarding consistency and asymptotic efficiency. On the other hand, there are currently only a handful articles on Bayesian single index models. In a Bayesian setup, we need a suitable prior process for the unknown nonparametric univariate function and another prior distribution on the sphere for the unknown coefficients. While using these types of priors, the posterior distribution of the coefficients does not have a closed form expression, and existing tools typically results in slow mixing of the Markov chain. In this article, we propose an efficient variant of the Metropolis Hastings algorithm to sample from the full conditional distribution using an Orsntein Uhlenbeck (OH) process for the nonparametric effect and a carefully chosen proposal density based on model alignment. The use of OU process results in fast convergence of the Markov chain.
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