Date of Award
Doctor of Philosophy
Daniel J Nordman
We consider the direct Bayes modeling of points partitioning a finite interval and defining break points in regression (or knots or locations of basis functions of known shape), where the order statistics from independent and identically distributed uniform observations on the interval prescribe the prior distribution. Such modeling is natural and direct, is arguably as non-informative as possible, and enables extremely simple Metropolis steps in successive substitution Markov chain Monte Carlo (MCMC) schemes — that we will then say employ the Sequential Uniform Proposal (SUPr) algorithm. The approach opens a myriad of possible new flexible Bayesian regression methods. In particular, where the SUPr algorithm is applied to both a covariate and a parameter of a response distribution, effective Bayes smooth isotonic regression methods arise.
Here we introduce SUPr and demonstrate its use and performance by applications to both the covariate alone and then the covariate plus the response in a number of both real and artificial 1-predictor (non-parametric) regression problems. SUPr is applied to the covariate as well as the covariate plus the response, and its performance is compared to competing methods in the litera- ture. Next, we illustrate how the simplicity of the SUPr algorithm allows the SUPr framework to be readily extended and adapted to more general univariate regression contexts. Specifically, we illus- trate how simple alternatives to the uniform proposal distribution can reduce the sampling period for the MCMC scheme, how the ability to construct the response function allows us to directly esti- mate context-specific features of interest, and how the use of basis functions in regression problems produces highly flexible estimation. We then extend this univariate framework to a multivariate context, where SUPr serves as the computational engine for a projection-based kernel-weighted Bayesian regression model. Through simulation studies, we find that the SUPr methods compare favorably to other methods. We conclude that the framework provided by SUPr is an interesting and flexible Bayesian context in which many practically useful methods can be developed.
Mouzon, Ian, "The SUPr Algorithm" (2020). Graduate Theses and Dissertations. 18365.