VARIATIONAL INFERENCE FOR DIFFUSION MODU-LATED COX PROCESSES Anonymous

Abstract

This paper proposes a stochastic variational inference (SVI) method for computing an approximate posterior path measure of a Cox process. These processes are widely used in natural and physical sciences, engineering and operations research, and represent a non-trivial model of a wide array of phenomena. In our work, we model the stochastic intensity as the solution of a diffusion stochastic differential equation (SDE), and our objective is to infer the posterior, or smoothing, measure over the paths given Poisson process realizations. We first derive a system of stochastic partial differential equations (SPDE) for the pathwise smoothing posterior density function, a non-trivial result, since the standard solution of SPDEs typically involves an Itô stochastic integral, which is not defined pathwise. Next, we propose an SVI approach to approximating the solution of the system. We parametrize the class of approximate smoothing posteriors using a neural network, derive a lower bound on the evidence of the observed point process samplepath, and optimize the lower bound using stochastic gradient descent (SGD). We demonstrate the efficacy of our method on both synthetic and real-world problems, and demonstrate the advantage of the neural network solution over standard numerical solvers.

1. INTRODUCTION

Cox processes (Cox, 1955; Cox & Isham, 1980) , also known as doubly-stochastic Poisson processes, are a class of stochastic point processes wherein the point intensity is itself stochastic and, conditional on a realization of the intensity process, the number of points in any subset of space is Poisson distributed. These processes are widely used in the natural and physical sciences, engineering and operations research, and form useful models of a wide array of phenomena. We model the intensity by a diffusion process that is the solution of a stochastic differential equation (SDE). This is a standard assumption across a range of applications (Susemihl et al., 2011; Kutschireiter et al., 2020) . The measure induced by the solution of the SDE serves as a prior measure over sample paths, and our objective is to infer a posterior measure over the paths of the underlying intensity process, given realizations of the Poisson point process observations over a fixed time horizon. This type of inference problem has been studied in the Bayesian filtering literature (Schuppen, 1977; Bain & Crisan, 2008; Särkkä, 2013) , where it is of particular interest to infer the state of the intensity process at any past time given all count observations till the present time instant (the resulting posterior is called the smoothing posterior measure). In a seminal paper, Snyder (1972) derived a stochastic partial differential equation (SPDE) describing the dynamics of the corresponding posterior density for Cox processes. The solution of this smoothing SPDE requires the computation of an Itô stochastic integral with respect to the counting process. It has long been recognized (Clark, 1978; Davis, 1981; 1982) that for stochastic smoothing (and filtering) theory to be useful in practice, it should be possible to compute smoothing posteriors conditioned on a single observed sample path. However, Itô integrals are not defined pathwise and deriving a pathwise smoothing density is remarkably hard. 30 years after Synder's original work Elliott & Malcolm (2005) derived a pathwise smoothing SPDE in the form of a coupled system of forward and backward pathwise SPDEs. Nonetheless, solving the system of pathwise SPDEs, or sampling from the corresponding SDE, is still challenging and intractable in general. It is well known, for example, that numerical techniques for solving these SPDEs, such as the finite element

