Two-Stage Metropolis Hastings; Bayesian Conditional Density Estimation & Survival Analysis via Partition Modeling, Laplace Approximations, and Efficient Computation

Abstract

Bayesian statistical methods are known for their flexibility in modeling. This flexibility is possible because parameters can often be estimated via Markov chain Monte Carlo methods. In large datasets or models with many parameters, Markov chain Monte Carlo methods are insufficient and inefficient. We introduce the two-stage Metropolis-Hastings algorithm which modifies the proposal distribution of the Metropolis-Hastings algorithm via a screening stage to reduce the computational cost. The screening stage requires a cheap estimate of the log-likelihood and speeds up computation even in complex models such as Bayesian multivariate adaptive regression splines. Next, a partition model, constructed from a Voronoi tessellation, is proposed for conditional density estimation using logistic Gaussian processes. A Laplace approximation is used to approximate the marginal likelihood providing a tractable Markov chain Monte Carlo algorithm. In simulations and an application to windmill power output, the model successfully provides interpretation and flexibly models the densities. Last, a Bayesian tree partition model is proposed to model the hazard function of survival & reliability models. The piecewise-constant hazard function in each partition element is modeled via a latent Gaussian process. The marginal likelihood is estimated using Laplace approximations to yield a tractable reversible jump Markov chain Monte Carlo algorithm. The method is successful in simulations and provides insight into lung cancer survival rates in relation to protein expression levels.