Bayesian Multiscale Models for Poisson Processes

Article excerpt

1. INTRODUCTION

The focus of this article is a new class of models, Bayesian multi scale models (BMSM's), for phenomena with two defining characteristics: (1) the observed data may be well modeled as a discrete signal (time series) of independent Poisson counts, and (2) the underlying intensity function potentially has structure at multiple scales. Fundamental to the analysis of such data is the estimation of the underlying intensity function, and it is this problem in particular that is addressed. Knowledge of the intensity may be a goal in itself or may serve as an initial step prior to higher-level analyses (e.g., detection or classification).

1.1 A Motivating Example: Gamma-Ray Bursts

Phenomena of the sort just described are widespread throughout much of the experimental sciences. Examples include gamma-ray burst (GRB) signals in astronomy (e.g., Meegan et al. 1992); intensity transients in molecular spectroscopy (e.g., Vanden Bout et al. 1997); cross sections from emission-based modalities in medical imaging, and luminescence signals used in geological and archaeological dating (e.g., Duller, Li, Musson, and Wintle 1992).

Figure 1 shows two examples of the mysterious GRB signals encountered in the field of high-energy astrophysics. These data were collected by the BATSE (burst and transient source experiment) instruments on board NASA's Compton Gamma Ray Observatory (see Meegan et al. 1992). Operational since 1991, these instruments record the arrival times of gamma-ray photons corresponding to detected GRB's. The time series shown in Figure 1 are constructed, as is typical in practice, by aggregating the arrival times into intervals of equal length. Currently the BATSE instruments have recorded data on more than 4,000 bursts; the resulting time series have been compared to snowflakes in that "no two are alike." Widely described as one of modern astronomy's greatest mysteries, the physical mechanisms underlying GRB's still remain almost entirely unknown after almost 30 years of study! (However, see Wijers 1998, and accompanying articles, for a discussion of some recent breakthroughs, giving evidence that GRB's are some of the earliest and farthest observed phenomena in the universe.) Accordingly, fundamental tasks such as estimation and classification are critically important in the development of physical models at this stage. An analysis of the two bursts in Figure 1, using the Bayesian multiscale models introduced here, is presented in Section 6.

1.2 Organization of this Article

The rest of this article is organized as follows. As a first and fundamental step to the rest of the development, a multiscale factorization of the Poisson likelihood function is introduced in Section 2. This factorization is induced by a recursive dyadic partition of the data space, to which is connected a binary tree representation. In Section 3 the new class of Bayesian multiscale models is introduced. Specifically, a multiscale prior distribution is added to the factorization developed in Section 2, by placing mixture distributions at each node of the underlying binary tree. Then this basic model structure is embedded within a larger, richer framework through model mixing on a complete set of binary trees. This latter step is preceeded by an appropriate justification and followed by a brief exploration of the properties corresponding to the full model. Next, the Bayes optimal estimator is derived for the unknown intensity function in Section 4, under squared-error loss, and is found to possess a recursive structure that allows for an efficient computational strategy. Elicitation of the relevant hyperparameters is addressed in Section 5, where an empirical Bayes approach is outlined for eliciting data-dependent values for hyperparameters interpretable as the "fraction of homogeneity" at each scale in the underlying intensity. Practical performance characteristics of the method are explored in Section 6, using both simulations and the GRB data in Figure 1. …