Bayesian multiscale models for Poisson processes

I introduce a class of Bayesian multiscale models (BMSM's) for one-dimensional inhomogeneous Poisson processes. The focus is on estimating the (discretized) intensity function underlying the process. Unlike the usual transform-based approach at the heart of most wavelet-based methods for Gaussian data, these BMSM's are constructed using recursive dyadic partitions (RDP's) within an entirely likelihood-based framework. Each RDP may be associated with a binary tree, and a new multiscale prior distribution is introduced for the unknown intensity through the placement of mixture distributions at each of the nodes of the tree. The concept of model mixing is then applied to a complete collection of such trees. In addition to allowing for the inclusion of full location/scale information in the model, this last step also is fundamental both in inducing stationarity in the prior distribution and in enabling a given intensity function to be approximated at the resolution of the data. Under squared-error loss, a closed-form recursive expression for the Bayes optimal estimator is derived, which makes computationally efficient implementation possible. The mixing parameters in the prior distribution can be interpreted as the "fraction of homogeneity" in the underlying intensity function at each scale, and I provide an empirical Bayes approach to eliciting their values, resulting in the ability to quantify multiscale structure in the data. The practical performance of the overall procedure is investigated through a series of simulations and illustrated using a real-data example from the field of high-energy astrophysics.