Regeneration-enriched Markov processes with application to Monte Carlo
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We study a class of Markov processes comprising local dynamics governed by a fixed Markov process which are enriched with regenerations from a fixed distribution at a state-dependent rate. We give conditions under which such processes possess a given target distribution as their invariant measures, thus making them amenable for use within Monte Carlo methodologies. Enrichment imparts a number of desirable theoretical and methodological properties: Since the regeneration mechanism can compensate the choice of local dynamics, while retaining the same invariant distribution, great flexibility can be achieved in selecting local dynamics, and mathematical analysis is simplified. In addition we give straightforward conditions for the process to be uniformly ergodic and possess a coupling from the past construction that enables exact sampling from the invariant distribution. More broadly, the sampler can also be used as a recipe for introducing rejection-free moves into existing Markov Chain Monte Carlo samplers in continuous time.
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