Hierarchical Density Regression Using the Grid Mixture Model with Spatial Smoothing

• Activity Number: 312
• Type: Topic Contributed
• Date/Time: Tuesday, August 9, 2005 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Tyson Rogers*+ and Robert Weiss
• Companies: University of Minnesota and University of California, Los Angeles
• Keywords: Bayesian statistics ; hierarchical model ; density estimation ; flow cytometry ; HIV

Abstract

We develop a flexible parametric model for density estimation that we call the Grid Mixture Model (GMM). The GMM models an unknown density as a finite mixture of normal densities whose locations are fixed on a regularly spaced grid. The GMM's flexibility arises from including a large number of components in the finite mixture. The mixture weight and covariance matrix of each component is estimated given data sampled from the unknown density. We incorporate a conditionally autoregressive spatial prior for the mixture weights to model the expectation that components located near one another on the grid have similar weights. We then use the GMM in a hierarchical model to perform density regression wherein multiple unknown densities are estimated simultaneously. Each unknown density has a corresponding set of covariates. The hierarchical model relates the covariates to the shape of the density function with a linear model for the mixture weights on the logit scale. We illustrate the model using samples of flow cytometry data collected on a set of HIV+ and HIV- individuals.