  Many probabilistic inference problems such as stochastic filtering or the
computation of rare event probabilities require model analysis under initial
and terminal constraints. We propose a solution to this bridging problem for
the widely used class of population-structured Markov jump processes. The
method is based on a state-space lumping scheme that aggregates states in a
grid structure. The resulting approximate bridging distribution is used to
iteratively refine relevant and truncate irrelevant parts of the state-space.
This way the algorithm learns a well-justified finite-state projection yielding
guaranteed lower bounds for the system behavior under endpoint constraints. We
demonstrate the method's applicability to a wide range of problems such as
Bayesian inference and the analysis of rare events.
