stratified MCMC

When working last week with a student, we came across [the slides of a talk at ICERM by Brian van Koten about] a stratified MCMC method whose core idea is to solve a eigenvector equation z’=z’F associated with the masses of “partition” functions Ψ evaluated at the target. (The arXived paper is also available since 2017 but I did not check it in more details.)Although the “partition” functions need to overlap for the matrix not to be diagonal (actually the only case that does not work is when these functions are truly indicator functions). As in other forms of stratified sampling, the practical difficulty is in picking the functions Ψ so that the evaluation of the terms of the matrix F is not overly impacted by the Monte Carlo error. If spending too much time in estimating these terms, there is not a clear gain in switching to stratified sampling, which may be why it is not particularly developed in the MCMC literature….

As an interesting aside, the illustration in this talk comes from the Mexican stamp thickness data I also used in my earlier mixture papers, concerning the 1872 Hidalgo issue that was printed on different qualities of paper. This makes the number k of components somewhat uncertain, although k=3 is sometimes used as a default. Hence a parameter and simulation space of dimension 8, even though the method is used toward approximating the marginal posteriors on the weights λ¹ and λ².

oops, no tooltip…!

Thanks to Jean-Michel Marin, I have just noticed that the legends I put on my (or others‘) images were not visible as blue boxes hovering over the picture. I checked on wordpress and found the solution: The new title of the downloaded image is not the title of the published image!

The title added in this field does not display as a tooltip (when a mouse is hovered over the image). To add a tooltip title, edit the image after inserting it into the post, and add your tooltip wording to the Title field.

However that means that the older images in the ‘Og (those published since the new version of wordpress was implemented, around the very end of November…) will not be corrected…

estimating the measure and hence the constant

As mentioned on my post about the final day of the ICERM workshop, Xiao-Li Meng addresses this issue of “estimating the constant” in his talk. It is even his central theme. Here are his (2011) slides as he sent them to me (with permission to post them!):

He therefore points out in slide #5 why the likelihood cannot be expressed in terms of the normalising constant because this is not a free parameter. Right! His explanation for the approximation of the unknown constant is then to replace the known but intractable dominating measure—in the sense that it cannot compute the integral—with a discrete (or non-parametric) measure supported by the sample. Because the measure is defined up to a constant, this leads to sample weights being proportional to the inverse density. Of course, this representation of the problem is open to criticism: why focus only on measures supported by the sample? The fact that it is the MLE is used as an argument in Xiao-Li’s talk, but this can alternatively be seen as a drawback: I remember reviewing Dankmar Böhning’s Computer-Assisted Analysis of Mixtures and being horrified when discovering this feature! I am currently more agnostic since this appears as an alternative version of empirical likelihood. There are still questions about the measure estimation principle: for instance, when handling several samples from several distributions, why should they all contribute to a single estimate of μ rather than to a product of measures? (Maybe because their models are all dominated by the same measure μ.) Now, getting back to my earlier remark, and as a possible answer to Larry’s quesiton, there could well be a Bayesian version of the above, avoiding the rough empirical likelihood via Gaussian or Drichlet process prior modelling.

ICERM, Brown, Providence, RI (#3)

Just yet another perfect day in Providence! Especially when I thought it was going to be a half-day: After a longer and slightly warmer run in the early morning around the peninsula, I attended the lecture by Eric Moulines on his recent results on adaptive MCMC and the equi-energy sampler. At this point, we were told that, since Peter Glynn was sick, the afternoon talks were drifted forward. This meant that I could attend Mylène Bédard’s talk in the morning and most of Xiao-Li Meng’s talk, before catching my bus to the airport, making it a full day in the end!

The research presented by Mylène (and coauthored with Randal Douc and Eric Moulines) was on multiple-try MCMC and delayed-rejection MCMC, with optimal scaling results and a comparison of the efficiency of those involved schemes. I had not seen the work before and got quite impressed by the precision of the results and the potential for huge efficiency gains. One of the most interesting tricks was to use an antithetic move for the second step, considerably improving the acceptance rate in the process. An aside exciting point was to realise that the hit-and-run solution was also open to wide time-savings thanks to some factorisation.

While Xiao-Li’s talk had connections with his earlier illuminating talk in New York last year, I am quite desolate to have missed [the most novel] half of it (and still caught my bus by a two minute margin!), esp. because it connected beautifully with the constant estimation controverse! Indeed, Xiao-Li started his presentation with the pseudo-paradox that the likelihood cannot be written as a function of the normalising constant, simply because this is not a free parameter. He then switched to his usual theme that the dominating measure was to be replaced with a substitute and estimated.The normalising constant being a function of the dominating measure, it is a by-product of this estimation step. And can even be endowed within a Bayesian framework. Obviously, one can always argue against the fact that the dominating measure is truly unknown, however this gives a very elegant safe-conduct to escape the debate about the constant that did not want to be estimated…So to answer Xiao-Li’s question as I was leaving the conference room, I have now come to a more complete agreement with his approach. And think further advances could be contemplated along this path…

ICERM, Brown, Providence, RI (#2)

Just another perfect day in Providence! After a brisk run in the eearly morning which took me through Brown campus, I attended the lecture by Sean Meyn on feedback particle filters. As it was mostly on diffusions with control terms, just too far from my field, I missed most of the points. (My fault, not Sean’s!) Then Ramon von Handel gave a talk about the curse(s) of dimensionality in particle filters, much closer to my interests, with a good summary of why (optimal) filters were not suffering from a curse in n, the horizon size, but in d, the dimension of the space, followed by an argument that some degree of correlation decay could overcome this dimensional curse as well. After the lunch break (where I thought further about the likelihood principle!), Dana Randall gave a technical talk on mixing properties of the hardcore model on Z² and bounding the cutoff parameter, which is when I appreciated the ability to follow talks from the ICERM lounge, watching slides and video of the talk taking place on the other side of the wall! At last, and in a programming contrapoint from slowly mixing to fastest mixing, Jim Fill presented his recent work on ordering Markov chains and finding fastest-mixing chains, which of course reminded me of Peskun ordering although there may be little connection in the end. The poster session in the evening had sufficiently few posters to make the discussion with each author enjoyable and relevant.A consistent feature of the meeting thus, allowing for quality interacting time between participants. I am now looking forward the final day with a most intriguing title by my friend Eric Moulines on TBA…