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How to get Metropolis-Hastings acceptance rates close to 1?
How come it’s possible to get Metropolis-Hastings acceptance rates close to 1 (for example, when exploring a unimodal distribution with a normal proposal distribution with too-small SD), after burn-in is over? I see it in my own MCMC chains but I don’t understand how it makes sense.
How is the Metropolis-Hastings algorithm used in statistics?
In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult.
When to use Metropolis-Hastings instead of MCMC?
Metropolis–Hastings, along with other MCMC methods, do not have this problem to such a degree, and thus are often the only solutions available when the number of dimensions of the distribution to be sampled is high.
How can autocorrelation be reduced in the Metropolis algorithm?
Autocorrelation can be reduced by increasing the jumping width (the average size of a jump, which is related to the variance of the jumping distribution), but this will also increase the likelihood of rejection of the proposed jump.
Why is Metropolis-Hastings considered an optimisation algorithm?
A possible reason for your confusion is the potential perception of the Metropolis-Hastings algorithm as an optimisation algorithm. The algorithm spends more iterations on higher target regions but does not aim at the maximum.
Which is the best acceptance rate for Metropolis?
It can be shown that acceptance rates between 20% and 30% are optimal for typical applications (more on that here ). Finally, we can plot the results. There are more elegant ways of plotting this which I discuss in another recent post, so check this out, but for the moment I didn’t want to use any packages, so we do it the hard way:
What’s the standard acceptance rate for a proposal?
As a side note, standard practice is to tune the proposal distribution to get around a 0.2-0.25 acceptance rate. See here for a discussion of this. While this sounds like an unrealistic example, a genuine illustration is the Gibbs sampler, which can be interpreted as a sequence of Metropolis-Hastings steps, all with probability one.