Abstract
We consider nonparametric Bayesian estimation and prediction for nonhomogeneous Poisson process models with unknown intensity functions. We propose a class of improper priors for intensity functions. Nonparametric Bayesian inference with kernel mixture based on the class improper priors is shown to be useful, although improper priors have not been widely used for nonparametric Bayes problems. Several theorems corresponding to those for finite-dimensional independent Poisson models hold for nonhomogeneous Poisson process models with infinite-dimensional parameter spaces. Bayesian estimation and prediction based on the improper priors are shown to be admissible under the Kullback-Leibler loss. Numerical methods for Bayesian inference based on the priors are investigated.
Highlights
We consider nonparametric Bayesian estimation and prediction for nonhomogeneous Poisson process models with unknown intensity functions
We show that the framework of statistical decision theory is effectively applied to prediction and estimation for nonhomogeneous Poisson process models with infinite dimensional parameter spaces
Few studies on admissibility concerning infinite-dimensional models have been carried out, we show that Blyth’s method with a convex loss is useful for our infinite-dimensional problem because the method works even when the support of the prior is a small subset of the whole parameter space
Summary
S TATISTICAL modeling and data analysis based on nonhomogeneous Poisson point processes have various applications (e.g. [31], [33]). Bayesian prediction based on shrinkage priors for finite-dimensional models such as the multivariate. | x) based on the prior πS(μ) = μ−(d−2) = μi2)−(d−2)/2 introduced by [32] dominates the Bayesian predictive density pJ(y | x) based on the Jeffreys prior πJ(μ) = 1 ( [20]). Theorem 2 ([21]): For every d ≥ 1, the Bayesian predictive densities based on the priors in the class {πα,γ(λ) : 0
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