optimisation via slice sampling

This morning, over breakfast; I read the paper recently arXived by John Birge et Nick Polson. I was intrigued by the combination of optimisation and of slice sampling, but got a wee disappointed by the paper, in that it proposes a simple form of simulated annealing, minimising f(x) by targeting a small collection of energy functions exp{-τf(x)}. Indeed, the slice sampler is used to explore each of those targets, i.e. for a fixed temperature τ. For the four functions considered in the paper, a slice sampler can indeed be implemented, but this feature could be seen as a marginalia, given that a random walk Metropolis-Hastings algorithm could be used as a proposal mechanism in other cases. The other intriguing fact is that annealing is not used in the traditional way of increasing the coefficient τ along iterations (as in our SAME algorithm), for which convergence issues are much more intricate, but instead stays at the level where a whole (Markov) sample is used for each temperature, the outcomes being then compared in terms of localisation (and maybe for starting at the next temperature value). I did not see any discussion about the selection of the successive temperatures, which usually is a delicate issue in realistic settings, nor of the stopping rule(s) used to decide the maximum has been reached.

ABC-MCMC for parallel tempering

“In this paper a new algorithm combining population-based MCMC methods with ABC requirements is proposed, using an analogy with the Parallel Tempering algorithm (Geyer, 1991).“

Another of those arXiv papers that had sat on my to-read pile for way too long: Likelihood-free parallel tempering by Meïli Baragtti, Agnès Grimaud, and Denys Pommeret, from Luminy, Marseilles. The paper mentions our population Monte Carlo (PMC) algorithm (Beaumont et al., 2009) and other ABC-SMC algorithms, but opts instead for an ABC-MCMC basis. The purpose is to build a parallel tempering method. Tolerances and temperatures evolve simultaneously. I however fail to see where the tempering occurs in the algorithm (page 7): there is a set of temperatures T₁,….,T_N, but they do not appear within the algorithm. My first idea of a tempering mechanism in a likelihood-free setting was to replicate our SAME algorithm (Doucet, Godsill, and Robert, 2004), by creating T_j copies of the [pseudo-]observations to mimic the likelihood taken to the power T_j. But this is annealing, not tempering, and I cannot think of the opposite of copies of the data. Unless of course a power of the likelihood can be simulated (and even then, what would the equivalent be for the data…?) Maybe a natural solution would be to operate some kind of data-attrition, e.g. by subsampling the original vector of observations.

Discussing the issue with Jean-Michel Marin, during a visit to Montpellier today, I realised that the true tempering came from the tolerances ε_i, while the temperatures T_j were there to calibrate the proposal distributions. And that the major innovation contained in the thesis (if not so clearly in the paper) was to boost exchanges between different tolerances, improving upon the regular ABC-MCMC sampler by an equi-energy move.

Feedback on data cloning

Following some discussions I had last week at Banff about data cloning, I re-read the 2007 “Data cloning” paper published in Ecology Letters by Lele, Dennis, and Lutscher. Once again, I see a strong similarity with our 2002 Statistics and Computing SAME algorithm, as well as with the subsequent (and equally similar) “A multiple-imputation Metropolis version of the EM algorithm” published in Biometrika by Gaetan and Yao in 2003—Biometrika to which Arnaud and I had earlier and unsuccessfully submitted this unpublished technical report on the convergence of the SAME algorithm… (The SAME algorithm is also described in detail in the 2005 book Inference in Hidden Markov Models, Chapter 13.)

10w2170, Banff [2]

Over the two days of the Hierarchical Bayesian Methods in Ecology workshop, we managed to cover normal models, testing, regression, Gibbs sampling, generalised linear models, Metropolis-Hastings algorithms and of course a fair dose of hierarchical modelling. At the end of the Saturday marathon session, we spent one and half discussing some models studied by the participants, which were obviously too complex to be solved on the spot but well-defined so that we could work on MCMC implementation and analysis. And on Sunday morning, a good example of Poisson regression proposed by Devin Goodman led to an exciting on-line programming of a random effect generalised model, with the lucky occurrence of detectable identifiability issues that we could play with… I am impressed at the resilience of the audience given the gruesome pace I pursued over those two days, covering the five first chapters of Bayesian Core, all the way to the mixtures! In retrospect, I think I need to improve my coverage of testing as the noninformative case presumably sounded messy. And unconvincing. I also fear the material on hierarchical models was not sufficiently developed. But, overall, the workshop provided a wonderful opportunity to exchange with bright PhD students from Ecology and Forestry about their models and (hierarchical) Bayesian modelling.