**T**he title of this recent arXival had potential appeal, however the proposal ends up being rather straightforward and hence anti-climactic! The paper by Hu, Hendry and Heng proposes to run a mixture of proposals centred at the various modes of the target for an efficient exploration. This is a correct MCMC algorithm, granted!, but the requirement to know beforehand *all* the modes to be explored is self-defeating, since the major issue with MCMC is about modes that are omitted from the exploration and remain undetected throughout the simulation… As provided, this is a standard MCMC algorithm with no adaptive feature and I would rather suggest our population Monte Carlo version, given the available information. Another connection with population Monte Carlo is that I think the performances would improve by Rao-Blackwellising the acceptance rate, i.e. removing the conditioning on the (ancillary) component of the index. For PMC we proved that using the mixture proposal in the ratio led to an ideally minimal variance estimate and I do not see why randomising the acceptance ratio in the current case would bring any improvement.

## Archive for Rao-Blackwellisation

## efficient exploration of multi-modal posterior distributions

Posted in Books, Statistics, University life with tags acceptance probability, Metropolis-Hastings algorithms, multimodal target, population Monte Carlo, Rao-Blackwellisation on September 1, 2014 by xi'an## recycling accept-reject rejections

Posted in Statistics, University life with tags accept-reject algorithm, arXiv, auxiliary variable, Data augmentation, George Casella, intractable likelihood, Monte Carlo Statistical Methods, Rao-Blackwellisation, recycling, untractable normalizing constant on July 1, 2014 by xi'an**V**inayak Rao, Lizhen Lin and David Dunson just arXived a paper which proposes anew technique to handle intractable normalising constants. And which exact title is Data augmentation for models based on rejection sampling. (Paper that I read in the morning plane to B’ham, since this is one of my weeks in Warwick.) The central idea therein is that, if the sample density (*aka* likelihood) satisfies

where all terms but p are known in closed form, then completion by the rejected values of an hypothetical accept-reject algorithm−hypothetical in the sense that the data does not have to be produced by an accept-reject scheme but simply the above domination condition to hold−allows for a data augmentation scheme. Without requiring the missing normalising constant. Since the completed likelihood is

A closed-form, if not necessarily congenial, function.

**N**ow this is quite a different use of the “rejected values” from the accept reject algorithm when compared with our 1996 Biometrika paper on the Rao-Blackwellisation of accept-reject schemes (which, still, could have been mentioned there… Or Section 4.2 of Monte Carlo Statistical Methods. Rather than re-deriving the joint density of the augmented sample, “accepted+rejected”.)

**I**t is a neat idea in that it completely bypasses the approximation of the normalising constant. And avoids the somewhat delicate tuning of the auxiliary solution of Moller et al. (2006) The difficulty with this algorithm is however in finding an upper bound M on the unnormalised density f that is

- in closed form;
- with a manageable and tight enough “constant” M;
- compatible with running a posterior simulation conditional on the added rejections.

The paper seems to assume further that the bound M is independent from the current parameter value θ, at least as suggested by the notation (and Theorem 2), but this is not in the least necessary for the validation of the formal algorithm. Such a constraint would pull M higher, hence reducing the efficiency of the method. Actually the matrix Langevin distribution considered in the first example involves a bound that depends on the parameter κ.

**T**he paper includes a result (Theorem 2) on the uniform ergodicity that relies on heavy assumptions on the proposal distribution. And a rather surprising one, namely that the probability of *rejection* is bounded from below, i.e. calling for a *less* efficient proposal. Now it seems to me that a uniform ergodicity result holds as well when the probability of *acceptance* is bounded from below since, then, the event when no rejection occurs constitutes an atom from the augmented Markov chain viewpoint. There therefore occurs a renewal each time the rejected variable set ϒ is empty, and ergodicity ensues (Robert, 1995, *Statistical Science*).

**N**ote also that, despite the opposition raised by the authors, the method *per se* does constitute a pseudo-marginal technique à la Andrieu-Roberts (2009) since the independent completion by the (pseudo) rejected variables produces an unbiased estimator of the likelihood. It would thus be of interest to see how the recent evaluation tools of Andrieu and Vihola can assess the loss in efficiency induced by this estimation of the likelihood.

*Maybe some further experimental evidence tomorrow…*

## Bayesian multimodel inference by RJMCMC: A Gibbs sampling approach

Posted in Books, pictures, Statistics, Travel, University life with tags Dunedin, Gibbs sampling, model comparison, New Zealand, palette, Rao-Blackwellisation, RJMCMC on December 27, 2013 by xi'an**B**arker (from the lovely city of Dunedin) and Link published a paper in the American Statistician last September that I missed, as I missed their earlier email about the paper since it arrived The Day After… The paper is about a new specification of RJMCMC, almost twenty years after Peter Green’s (1995) introduction of the method. The authors use the notion of *a palette*, “from which all model specific parameters can be calculated” (in a deterministic way). One can see the palette ψ as an intermediary step in the move between two models. This reduces the number of bijections, if not the construction of the dreaded Jacobians!, but forces the construction of pseudo-priors on the unessential parts of ψ for *every* model. Because the dimension of ψ is fixed, a Gibbs sampling interleaving model index and palette value is then implementable. The conditional of the model index given the palette is available provided there are not too many models under competitions, with the probabilities recyclable towards a Rao-Blackwell approximation of the model probability. I wonder at whether or not another Rao-Blackwellisation is possible, namely to draw from all the simulated palettes a sample for the parameter of an arbitrarily chosen model.

## On the use of marginal posteriors in marginal likelihood estimation via importance-sampling

Posted in R, Statistics, University life with tags Bayes factor, Chib's approximation, evidence, harmonic mean estimator, label switching, latent variable, marginal likelihood, MCMC, mixtures, Monte Carlo Statistical Methods, nested sampling, Poisson regression, Rao-Blackwellisation, simulation on November 20, 2013 by xi'an**P**errakis, Ntzoufras, and Tsionas just arXived a paper on marginal likelihood (evidence) approximation (with the above title). The idea behind the paper is to base importance sampling for the evidence on simulations from the product of the (block) marginal posterior distributions. Those simulations can be directly derived from an MCMC output by randomly permuting the components. The only critical issue is to find good approximations to the marginal posterior densities. This is handled in the paper either by normal approximations or by Rao-Blackwell estimates. the latter being rather costly since one importance weight involves B.L computations, where B is the number of blocks and L the number of samples used in the Rao-Blackwell estimates. The time factor does not seem to be included in the comparison studies run by the authors, although it would seem necessary when comparing scenarii.

**A**fter a standard regression example (that did not include Chib’s solution in the comparison), the paper considers 2- and 3-component mixtures. The discussion centres around label switching (of course) and the deficiencies of Chib’s solution against the current method and Neal’s reference. The study does not include averaging Chib’s solution over permutations as in Berkoff et al. (2003) and Marin et al. (2005), an approach that does eliminate the bias. Especially for a small number of components. Instead, the authors stick to the log(k!) correction, despite it being known for being quite unreliable (depending on the amount of overlap between modes). The final example is Diggle et al. (1995) longitudinal Poisson regression with random effects on epileptic patients. The appeal of this model is the unavailability of the integrated likelihood which implies either estimating it by Rao-Blackwellisation or including the 58 latent variables in the analysis. (There is no comparison with other methods.)

**A**s a side note, among the many references provided by this paper, I did not find trace of Skilling’s nested sampling or of safe harmonic means (as exposed in our own survey on the topic).

## another vanilla Rao-Blackwellisation

Posted in Statistics, University life with tags importance sampling, MCMC, Monte Carlos Statistical Methods, parallel processing, Rao-Blackwellisation, simulation on September 16, 2013 by xi'an**I**n the latest issue of ** Statistics and Computing** (2013, Issue 23, pages 577-587), Iliopoulos and Malefaki published a paper that relates to our vanilla Rao-Blackwellisation paper with Randal Douc. The idea is to derive another approximation to the ideal importance sampling weight using the “accepted” Markov chain. (With Randal, we had a Bernoulli factory representation.) The density g(x) of the accepted chain being unknown; it is represented as the expectation under π of the function

and hence estimated by a self-normalised average based on the whole Markov chain. This means the resulting importance estimate uses twice the output of the algorithm and that it is biased and of order O(n²), thus the same order as our original Rao-Blackwellised estimator (Robert & Casella, 1996)… This also means convergence and CLT are very hard to establish: the main convergence theorem of the paper holds only for finite state spaces, with a surprising smaller asymptotic variance for this self-normalised average than for the ideal importance sampling estimator in the independent Metropolis-Hastings case. (Both are biased by being self-normalised and the paper does not consider the magnitude of those biases.)

**I**nterestingly, the authors also ran a comparison with our parallelised Rao-Blackwellised version (with Pierre Jacob and Murray Smith), but conclude (P.58) at a larger CPU (should be GPU!!) required by the parallelisation, which does not really make sense: when compared with the plain Metropolis-Hastings implementation, run on a single processor, the parallel version only requires an extra random permutation per thread or per processor. I thus suspect a faulty implementation that induces this CPU being linear in the size of the blocks, like maybe only saving one output per block… Also interestingly, the paper re-analyses the Pima Indian probit model Jean-Michel Marin and I (and many others) used as benchmark in several of our papers. As in the most standard examples, the outcome shows a mild reduction in variance when using this estimated importance sampling version. Maybe a comparison with the ideal importance sampler (i.e. the one that does not divide by the sum of the weights since using normalised versions of the target and importance densities) would have helped in the comparison.

## snapshot from Budapest (EMS 2013 #3)

Posted in pictures, Statistics, University life, Wines with tags Bayesian statistics, Budapest, Hungary, machine learning, Markov chains, Rao-Blackwellisation on July 25, 2013 by xi'an**H**ere is the new version of the talk:

And I had a fairly interesting day at the conference, from Randal’s talk on hidden Markov models with finite valued observables to the two Terrys invited session (Terry Lyons vs. Terry Speed) to the machine learning session organised by a certain Michal Jordan (on the program) that turned out to be Michael Jordan (with a talk on the fusion between statistics and computability). A post-session chat with Terry Lyons also opened (to me) new perspectives on data summarisation. (And we at last managed to get a convergence result using a Rao-Blackwellisation argument!) Plus, we ended up the day in a nice bistrot called Zeller with an awfully friendly staff cooking family produces and serving fruity family wines and not yet spoiled by being ranked #1 on tripadvisor (but visibly attracting a lot of tourists like us).