The probabilistic programming paradigm is gaining popularity due to the possibility of easily representing probabilistic systems and running a number of off-the-shelf inference algorithms on them. This paper explores how this paradigm can be used to analyse collective systems, in the form of Markov Population Processes (MPPs). MPPs have been extensively used to represent systems of interacting agents, but their analysis is challenging due to the high computational cost required to perform exact simulations of the systems. We represent MPPs as runs of the approximate variant of the Stochastic Simulation Algorithm (SSA), known as $\tau$-leaping, which can be seen as a probabilistic program. We apply Gaussian Semantics, a recently proposed inference method for probabilistic programs, to analyse it. We show that $\tau$-leaping runs can be effectively analysed using a tailored version of Second Order Gaussian Approximation in which we use a Gaussian Mixture encoding of Poisson distributions. In the resulting analysis, the state of the system is approximated by a multivariate Gaussian Mixture generalizing other common Gaussian approximations such as the Linear Noise Approximation and the Langevin Method. Preliminary numerical experiments show that this approach is able to analyse MPPs with reasonable accuracy on the significant statistics while avoiding expensive numerical simulations.
