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I was reading Christian Robert's Blog today and quite liked the new Metropolis-Hastings algorithm he was discussing. It seemed simple and easy to implement.

Whenever I code up MCMC, I tend to stick with very basic MH algorithms, such as independent moves or random walks on the log scale.

What MH algorithms do people routinely use? In particular:

  • Why do you use them?
  • In some sense you must think that they are optimal - after all you use them routinely! So how do you judge optimality: ease-of-coding, convergence, ...

I'm particularly interested in what is used in practice, i.e. when you code up your own schemes.

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  • $\begingroup$ Perhaps, CW? The question seems like a poll of what people use. How would you define the 'best' answer? I admit that I am a bit fuzzy when to apply CW. So, feel free to ignore this comment if you feel otherwise. $\endgroup$
    – user28
    Sep 3 '10 at 15:22
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    $\begingroup$ I wouldn't mind leaving this as non-CW, especially if Colin can rephrase it slightly to allow for the possibility of one best answer. That said, I can't envision how to do that... $\endgroup$
    – Shane
    Sep 3 '10 at 16:05
  • $\begingroup$ I've tried to change the question to make it less CW - not sure if I have succeeded :( @Shane @Srikant if you still think that it should be a CW, feel free to change it. $\endgroup$ Sep 4 '10 at 10:50
  • $\begingroup$ This sounds like it would make a good survey paper! $\endgroup$
    – Sean
    Jun 7 '14 at 5:36
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Hybrid Monte Carlo is the standard algorithm used for neural networks. Gibbs sampling for Gaussian process classification (when not using a deterministic approximation instead).

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MH sampling is used when it's difficult to sample from the target distribution (e.g., when the prior isn't conjugate to the likelihood). So you use a proposal distribution to generate samples and accept/reject them based on the acceptance probability. The Gibbs sampling algorithm is a particular instance of MH where the proposals are always accepted. Gibbs sampling is one of the most commonly used algorithm due to its simplicity but it may not always to possible to apply, in which case one resorts to MH based on accept/reject proposals.

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In physics, statistical physics in particular, Metropolis-type algorithm(s) are used extensively. There are really countless variants of these, and the new ones are being actively developed. It's much too broad topic to give any sort of expanation here, so if you're interested you can start e.g. from these lecture notes or from the ALPS library webpage (http://alps.comp-phys.org/mediawiki).

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  • $\begingroup$ I realise that there are countless variants of this algorithm. What I was interested in was which ones do people routinely use. $\endgroup$ Sep 6 '10 at 7:50
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I use a slice sampler - originally proposed by Neal(2003), which I tune through heuristic optimization.

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