Some Bayesian methods for univariate density estimation
Kathleen Rey · Iowa State University Digital Repository (Iowa State University) · 2018
Density estimation is a standard tool for investigating attributes of a continuous distribution assumed to have generated a finite sample of data. Performing density estimation in a Bayesian framework allows for prior information about the underlying distribution to be used in the estimation. However, unless the size of the sample is very small, it is typically not desirable for the prior information to overwhelm information provided by the sample. Under consideration are Bayes methods for univariate density estimators on (0,1), where prior information does not dominate the outcome, and the resulting estimates are flexible and not constrained to belong to any standard parametric class of densities. The first method, referred to as DUOS, is based on a Distribution of Uniform Order Statistics prior. DUOS uses a class of step functions with a prior on the bin endpoints to allow for random bin widths and locations. The second method is called GOLD which applies a Gaussian Process on a pre-Log-Density to a class of continuous functions. A full investigation of each Bayes density estimator is presented, including the use of Markov Chain Monte Carlo to simulate from the posterior distributions, inference on the results, and extensive simulation studies establishing their competitiveness with other methods. These simulation studies also functioned as a guide to the development of defaults for the prior parameters. Finally, the package biRd in R (R Core Team (2017)) provides the functions necessary to implement both methods and completely assess the Bayesian estimates of the density, the cumulative distribution function, and a variety of statistical summaries of the density estimate.