## Parallel processing of independent Metropolis-Hastings algorithms

**W**ith Pierre Jacob, my PhD student, and Murray Smith, from National Institute of Water and Atmospheric Research, Wellington, who actually started us on this project at the last and latest Valencia meeting, we have completed a paper on using parallel computing in independent Metropolis-Hastings algorithms. The paper is arXived and the abstract goes as follows:

In this paper, we consider the implications of the fact that parallel raw-power can be exploited by a generic Metropolis–Hastings algorithm if the proposed values are independent. In particular, we present improvements to the independent Metropolis–Hastings algorithm that significantly decrease the variance of any estimator derived from the MCMC output, for a null computing cost since those improvements are based on a fixed number of target density evaluations. Furthermore, the techniques developed in this paper do not jeopardize the Markovian convergence properties of the algorithm, since they are based on the Rao–Blackwell principles of Gelfand and Smith (1990), already exploited in Casella and Robert 91996), Atchadé and Perron (2005) and Douc and Robert (2010). We illustrate those improvement both on a toy normal example and on a classical probit regression model but insist on the fact that they are universally applicable.

I am quite excited about the results in this paper, which took advantage of (a) older works of mine on Rao-Blackwellisation, (b) Murray’s interests in costly likelihoods, and (c) our mutual excitement when hearing about GPU parallel possibilities from Chris Holmes’ talk in Valencia. (As well as directions drafted in an exciting session in Vancouver!) The (free) gains over standard independent Metropolis-Hastings estimates are equivalent to using importance sampling gains, while keeping the Markov structure of the original chain. Given that 100 or more parallel threads can be enhanced from current GPU cards, this is clearly a field with much potential! The graph below

gives the variance improvements brought by three Rao-Blackwell estimates taking advantage of parallelisation over the initial MCMC estimate (first entry) with the importance sampling estimate (last entry) using only 10 parallel threads.

January 23, 2012 at 12:13 am

[...] second paper studies some particular choices for the weights in a much less adaptive scheme (where parallelisation would be an appropriate alternative, since each proposal in the multiple try only depends on the [...]

August 2, 2011 at 10:53 am

[...] with parallel computing is rather formal, having worked with Pierre Jacob and Murray Smith on the valid parallelisation of Metropolis-Hastings algorithms, but it was interesting to hear of the multiplicity of available [...]

July 17, 2011 at 11:53 pm

[...] processing units (GPUs) are all the rage these days. Most journal issues would be incomplete if at least one article didn’t mention [...]

March 15, 2011 at 12:13 am

[...] have now completed our revision of the parallel computation paper and hope to send it to JCGS within a few days. As seen on the arXiv version, and given the very positive reviews we received, [...]

February 13, 2011 at 12:16 am

[...] have now received reports back from JCGS for our parallel MCMC paper and they all are very nice and supportive! The reviewers essentially [...]