Given a Bayesian network and evidence for the values of a subset of the variables, a standard question is to compute the posterior distribution on the remaining variables. The Gibbs sampling technique gives a discrete time Markov chain which samples from this posterior distribution. Is there a continuous time Markov process which does the same thing?

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For any discrete time Markov chain, you can get a continuous time Markov chain with the same stationary measure: just perform the same transitions as the discrete time Markov chain, but at times given by a Poisson process.