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?
$\begingroup$ Can you clarify how/why you modify the posterior? Do you obtain new data? $\endgroup$– Juho KokkalaOct 6, 2014 at 17:16
$\begingroup$ I observe more data point. Basically I start with a uniform prior, and I observe data points one by one. After observing a data point I update the probability distribution, and I want to calculate a statistic, for example the expectation. A simple example: throw the coin several times and after each throw calculate the expected value. $\endgroup$– Artem GrotovOct 7, 2014 at 10:56
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.