I have a posterior distribution from which I calculate some statistics using sampling, for example I calculate expectation. So I draw 1000 samples using Metropolis-Hastings and then I calculate their mean. After I calculate the statistic, I modify the posterior (using the Bayes law) and I want to recalculate the statistics again, this process repeats indefinitely. Now the question: can I reuse the samples somehow so that I do not have to regenerate all 1000 samples (too slow)? Can I do something smart so that when my posterior is a bit different I can use some of the old samples?
When doing M-H it's a good idea to propose from a distribution that is "similar" to the target distribution. Hence, you may try to use the generated samples from a previous simulation as proposals for the simulation of a new posterior that you consider to be "close / similar" to the previous one. You may sample with replacement the previous parameters values, maybe adding some "noise" to the sampled values, and use the independence sampler version of M-H.