Continuous-time Markov chains (CTMCs) can have combinatorial state spaces rendering the computation of transition probabilities, and hence probabilistic inference, difficult or impossible with existing methods. For countably infinite state spaces in particular, with the exception of some special cases, there is no general, consistent Monte Carlo approximation method available for computing transition probabilities or estimating parameters. We propose a particle-based Monte Carlo approach where sequences of states are imputed while holding times are marginalized. We demonstrate the performance of our method on both synthetic and real datasets, drawing from two important examples of CTMCs having combinatorial state spaces: string-valued mutation models in phylogenetics and nucleic acid folding pathways.
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