Expectation propagation for continuous time stochastic processes

Physical and technological processes frequently exhibit intrinsic stochasticity. The main mathematical framework to describe and reason about such systems is provided by the theory of continuous time (Markovian) stochastic processes. Such processes have been well studied in chemical physics for several decades as models of chemical reactions at very low concentrations [Gardiner, 1985, e.g.]. More recently, the theory has found novel and diverse areas of application including systems biology at the single cell level [Wilkinson, 2011], ecology [Volkov et al., 2007] and performance modelling in computer systems [Hillston, 2005], to name but a few. The popularity of the approach has been greatly enhanced by the availability of efficient and accurate simulation algorithms [Gillespie, 1977, Gillespie et al., 2013], which permit a numerical solution of medium-sized systems within a reasonable time frame. As with most of science, many of the application domains of continuous time stochastic processes are becoming increasingly data-rich, creating a critical demand for inference algorithms which can use data to calibrate the models and analyse the uncertainty in the predictions. This raises new challenges and opportunities for statistics and machine learning, and has motivated the development of several algorithms for efficient inference in these systems. In this paper, we focus on the Bayesian approach, and formulate the inverse problem in terms of obtaining an approximation to a posterior distribution over the stochastic process, given observations of the system and using existing scientific information to build a prior model of the process.