Computationally tractable approximate and smoothed Polya trees

A discrete approximation to the Polya tree prior suitable for latent data is proposed that enjoys surprisingly simple and efficient conjugate updating. This approximation is illustrated in two applied contexts: the implementation of a nonparametric meta-analysis involving studies on the relationship between alcohol consumption and breast cancer, and random intercept Poisson regression for Ache armadillo hunting treks. The discrete approximation is then smoothed with Gaussian kernels to provide a smooth density for use with continuous data; the smoothed approximation is illustrated on a classic dataset on galaxy velocities and on recent data involving breast cancer survival in Louisiana.