Abstract
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.
Highlights
Discrete-valued continuous-time Markov Jump Processes (MJP) are widely used to model the time evolution of complex discrete phenomena in continuous time
The state variables are counts of individual entities of different populations. Many tasks, such as the analysis of rare events or the inference of agent counts under partial observations naturally introduce terminal constraints on the system
Reachability analysis, which is relevant in the context of probabilistic verification [8,38], is a bridging problem where the endpoint constraint is the visit of a set of goal states
Summary
Discrete-valued continuous-time Markov Jump Processes (MJP) are widely used to model the time evolution of complex discrete phenomena in continuous time. Such problems naturally occur in a wide range of areas such as chemistry [16], systems biology [49,46], epidemiology [36] as well as queuing systems [10] and finance [39]. The state variables are counts of individual entities of different populations Many tasks, such as the analysis of rare events or the inference of agent counts under partial observations naturally introduce terminal constraints on the system. In these cases, the system’s initial state is known, as well as the system’s (partial) state at a later time-point. If the exact, full state of the process Xt has been observed at time 0 and T , the bridging distribution is given by
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