Recently, I've been questioning myself on the possibility of combining a discrete update and a continuous update on a single MCMC. Stephens (2000) in Algorithm 3.2 runs the process for a fixed amount of time and then use this result to update the rest while Mohammadi and Wit (2015) use a continuous update on one "parameter" and a discrete update on the other. Which is the correct way to go and why? I can't find any theoretical literature about it but I find it easier to discretize the continuous chain.
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$\begingroup$ Stephens (2000) does not mix a discrete and a continuous Markov chain, but rather simulates a continuous-time jump process where the duration in one state is connected with the target distribution. Just like the more general PDMP algorithms. $\endgroup$Xi'an– Xi'an2023-07-04 20:57:05 +00:00Commented Jul 4, 2023 at 20:57
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$\begingroup$ I was not aware of PDMP algorithms and certainly will have a look on those. I understand that Stephens does not actually mix the discrete and continuous chains which is why i was wondering if there is some work or some theory about it. $\endgroup$nico– nico2023-07-05 10:45:55 +00:00Commented Jul 5, 2023 at 10:45
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