  Advances in sampling schemes for Markov jump processes have recently enabled
multiple inferential tasks. However, in statistical and machine learning
applications, we often require that these continuous-time models find support
on structured and infinite state spaces. In these cases, exact sampling may
only be achieved by often inefficient particle filtering procedures, and
rapidly augmenting observed datasets remains a significant challenge. Here, we
build on the principles of uniformization and present a tractable framework to
address this problem, which greatly improves the efficiency of existing
state-of-the-art methods commonly used in small finite-state systems, and
further scales their use to infinite-state scenarios. We capitalize on the
marginal role of variable subsets in a model hierarchy during the process
jumps, and describe an algorithm that relies on measurable mappings between
pairs of states and carefully designed sets of synthetic jump observations. The
proposed method enables the efficient integration of slice sampling techniques
and it can overcome the existing computational bottleneck. We offer evidence by
means of experiments addressing inference and clustering tasks on both
simulated and real data sets.
