Unbiased Bayesian inference for population Markov jump processes via random truncations

We consider continuous time Markovian processes where populations of individual agents interact stochastically according to kinetic rules. Despite the increasing prominence of such models in fields ranging from biology to smart cities, Bayesian inference for such systems remains challenging, as these are continuous time, discrete state systems with potentially infinite state-space. Here we propose a novel efficient algorithm for joint state/parameter posterior sampling in population Markov Jump processes. We introduce a class of pseudo-marginal sampling algorithms based on a random truncation method which enables a principled treatment of infinite state spaces. Extensive evaluation on a number of benchmark models shows that this approach achieves considerable savings compared to state of the art methods, retaining accuracy and fast convergence. We also present results on a synthetic biology data set showing the potential for practical usefulness of our work.     

The entropic measure transform

We introduce the entropic measure transform (EMT) problem for a general process and prove the existence of a unique optimal measure characterizing the solution. The density process of the optimal measure is characterized using a semimartingale BSDE under general conditions. The EMT is used to reinterpret the conditional entropic risk-measure and to obtain a convenient formula for the conditional expectation of a process that admits an affine representation under a related measure. The EMT is then used to provide a new characterization of defaultable bond prices, forward prices and futures prices when a jump-diffusion drives the asset. The characterization of these pricing problems in terms of the EMT provides economic interpretations as maximizing the returns subject to a penalty for removing financial risk as expressed through the aggregate relative entropy. The EMT is shown to extend the optimal stochastic control characterization of default-free bond prices of Gombani & Runggaldier (2013). These methods are illustrated numerically with an example in the defaultable bond setting. The Canadian Journal of Statistics 48: 97–129; 2020 © 2020 Statistical Society of Canada