  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.
