Archive for reversible jump MCMC

normalising constants of G-Wishart densities

Posted in Books, Statistics with tags , , , , , , on June 28, 2017 by xi'an

Abdolreza Mohammadi, Hélène Massam, and Gérard Letac arXived last week a paper on a new approximation of the ratio of two normalising constants associated with two G-Wishart densities associated with different graphs G. The G-Wishart is the generalisation of the Wishart distribution by Alberto Roverato to the case when some entries of the matrix are equal to zero, which locations are associated with the graph G. While enjoying the same shape as the Wishart density, this generalisation does not enjoy a closed form normalising constant. Which leads to an intractable ratio of normalising constants when doing Bayesian model selection across different graphs.

Atay-Kayis and Massam (2005) expressed the ratio as a ratio of two expectations, and the current paper shows that this leads to an approximation of the ratio of normalising constants for a graph G against the graph G augmented by the edge e, equal to

Γ(½{δ+d}) / 2 √π Γ(½{δ+d+1})

where δ is the degree of freedom of the G-Wishart and d is the number of minimal paths of length 2 linking the two end points of e. This is remarkably concise and provides a fast approximation. (The proof is quite involved, by comparison.) Which can then be used in reversible jump MCMC. The difficulty is obviously in evaluating the impact of the approximation on the target density, as there is no manageable available alternative to calibrate the approximation. In a simulation example where such an alternative is available, the error is negligible though.

repulsive mixtures

Posted in Books, Statistics with tags , , , , , , , , on April 10, 2017 by xi'an

Fangzheng Xie and Yanxun Xu arXived today a paper on Bayesian repulsive modelling for mixtures. Not that Bayesian modelling is repulsive in any psychological sense, but rather that the components of the mixture are repulsive one against another. The device towards this repulsiveness is to add a penalty term to the original prior such that close means are penalised. (In the spirit of the sugar loaf with water drops represented on the cover of Bayesian Choice that we used in our pinball sampler, repulsiveness being there on the particles of a simulated sample and not on components.) Which means a prior assumption that close covariance matrices are of lesser importance. An interrogation I have has is was why empty components are not excluded as well, but this does not make too much sense in the Dirichlet process formulation of the current paper. And in the finite mixture version the Dirichlet prior on the weights has coefficients less than one.

The paper establishes consistency results for such repulsive priors, both for estimating the distribution itself and the number of components, K, under a collection of assumptions on the distribution, prior, and repulsiveness factors. While I have no mathematical issue with such results, I always wonder at their relevance for a given finite sample from a finite mixture in that they give an impression that the number of components is a perfectly estimable quantity, which it is not (in my opinion!) because of the fluid nature of mixture components and therefore the inevitable impact of prior modelling. (As Larry Wasserman would pound in, mixtures like tequila are evil and should likewise be avoided!)

The implementation of this modelling goes through a “block-collapsed” Gibbs sampler that exploits the latent variable representation (as in our early mixture paper with Jean Diebolt). Which includes the Old Faithful data as an illustration (for which a submission of ours was recently rejected for using too old datasets). And use the logarithm of the conditional predictive ordinate as  an assessment tool, which is a posterior predictive estimated by MCMC, using the data a second time for the fit.

SMC on a sequence of increasing dimension targets

Posted in Statistics with tags , , , , , , , , , on February 15, 2017 by xi'an

mixdirRichard Everitt and co-authors have arXived a preliminary version of a paper entitled Sequential Bayesian inference for mixture models and the coalescent using sequential Monte Carlo samplers with transformations. The central notion is an SMC version of the Carlin & Chib (1995) completion in the comparison of models in different dimensions. Namely to create auxiliary variables for each model in such a way that the dimension of the completed models are all the same. (Reversible jump MCMC à la Peter Green (1995) can also be interpreted this way, even though only relevant bits of the completion are used in the transitions.) I find the paper and the topic most interesting if only because it relates to earlier papers of us on population Monte Carlo. It also brought to my awareness the paper by Karagiannis and Andrieu (2013) on annealed reversible jump MCMC that I had missed at the time it appeared. The current paper exploits this annealed expansion in the devising of the moves. (Sequential Monte Carlo on a sequence of models with increasing dimension has been studied in the past.)

The way the SMC is described in the paper, namely, reweight-subsample-move, does not strike me as the most efficient as I would try to instead move-reweight-subsample, using a relevant move that incorporate the new model and hence enhance the chances of not rejecting.

One central application of the paper is mixture models with an unknown number of components. The SMC approach applied to this problem means creating a new component at each iteration t and moving the existing particles after adding the parameters of the new component. Since using the prior for this new part is unlikely to be at all efficient, a split move as in Richardson and Green (1997) can be considered, which brings back the dreaded Jacobian of RJMCMC into the picture! Here comes an interesting caveat of the method, namely that the split move forces a choice of the split component of the mixture. However, this does not appear as a strong difficulty, solved in the paper by auxiliary [index] variables, but possibly better solved by a mixture representation of the proposal, as in our PMC [population Monte Carlo] papers. Which also develop a family of SMC algorithms, incidentally. We found there that using a mixture representation of the proposal achieves a provable variance reduction.

“This puts a requirement on TSMC that the single transition it makes must be successful.”

As pointed by the authors, the transformation SMC they develop faces the drawback that a given model is only explored once in the algorithm, when moving to the next model. On principle, there would be nothing wrong in including regret steps, retracing earlier models in the light of the current one, since each step is an importance sampling step valid on its own right. But SMC also offers a natural albeit potentially high-varianced approximation to the marginal likelihood, which is quite appealing when comparing with an MCMC outcome. However, it would have been nice to see a comparison with alternative estimates of the marginal in the case of mixtures of distributions. I also wonder at the comparative performances of a dual approach that would be sequential in the number of observations as well, as in Chopin (2004) or our first population Monte Carlo paper (Cappé et al., 2005), since subsamples lead to tempered versions of the target and hence facilitate moves between models, being associated with flatter likelihoods.

Bayesian model selection without evidence

Posted in Books, Statistics, University life with tags , , , , , , , on September 20, 2016 by xi'an

“The new method circumvents the challenges associated with accurate evidence calculations by computing posterior odds ratios using Bayesian parameter estimation”

One paper leading to another, I had a look at Hee et al. 2015 paper on Bayes factor estimation. The “novelty” stands in introducing the model index as an extra parameter in a single model encompassing all models under comparison, the “new” parameterisation being in (θ,n) rather than in θ. With the distinction that the parameter θ is now made of the union of all parameters across all models. Which reminds us very much of Carlin and Chib (1995) approach to the problem. (Peter Green in his Biometrika (1995) paper on reversible jump MCMC uses instead a direct sum of parameter spaces.) The authors indeed suggest simulating jointly (θ,n) in an MCMC or nested sampling scheme. Rather than being updated by arbitrary transforms as in Carlin and Chib (1995) the useless parameters from the other models are kept constant… The goal being to estimate P(n|D) the marginal posterior on the model index, aka the posterior probability of model n.

Now, I am quite not certain keeping the other parameter constants is a valid move: given a uniform prior on n and an equally uniform proposal, the acceptance probability simplifies into the regular Metropolis-Hastings ratio for model n. Hence the move is valid within model n. If not, I presume the previous pair (θ⁰,n⁰) is repeated. Wait!, actually, this is slightly more elaborate: if a new value of n, m, is proposed, then the acceptance ratio involves the posteriors for both n⁰ and m, possibly only the likelihoods when the proposal is the prior. So the move will directly depend on the likelihood ratio in this simplified case, which indicates the scheme could be correct after all. Except that this neglects the measure theoretic subtleties that led to reversible jump symmetry and hence makes me wonder. In other words, it follows exactly the same pattern as reversible jump without the constraints of the latter… Free lunch,  anyone?!

reversible chain[saw] massacre

Posted in Books, pictures, R, Statistics, University life with tags , , , , , , , , on May 16, 2016 by xi'an

A paper in Nature this week that uses reversible-jump MCMC, phylogenetic trees, and Bayes factors. And that looks at institutionalised or ritual murders in Austronesian cultures. How better can it get?!

“by applying Bayesian phylogenetic methods (…) we find strong support for models in which human sacrifice stabilizes social stratification once stratification has arisen, and promotes a shift to strictly inherited class systems.” Joseph Watts et al.

The aim of the paper is to establish that societies with human sacrifices are more likely to have become stratified and stable than societies without such niceties. The hypothesis to be tested is then about the evolution towards more stratified societies rather the existence of a high level of stratification.

“The social control hypothesis predicts that human sacrifice (i) co-evolves with social stratification, (ii) increases the chance of a culture gaining social stratification, and (iii) reduces the chance of a culture losing social stratification once stratification has arisen.” Joseph Watts et al.

The methodological question is then how can this be tested when considering those are extinct societies about which little is known. Grouping together moderate and high stratification societies against egalitarian societies, the authors tested independence of both traits versus dependence, with a resulting Bayes factor of 3.78 in favour of the latest. Other hypotheses of a similar flavour led to Bayes factors in the same range. Which is thus not overwhelming. Actually, given that the models are quite simplistic, I do not agree that those Bayes factors prove anything of the magnitude of such anthropological conjectures. Even if the presence/absence of human sacrifices is confirmed in all of the 93 societies, and if the stratification of the cultures is free from uncertainties, the evolutionary part is rather involved, from my neophyte point of view: the evolutionary structure (reproduced above) is based on a sample of 4,200 trees based on Bayesian analysis of Austronesian basic vocabulary items, followed by a call to the BayesTrait software to infer about evolution patterns between stratification levels, concluding (with p-values!) at a phylogenetic structure of the data. BayesTrait was also instrumental in deriving MLEs for the various transition rates, “in order to inform our choice of priors” (!). BayesTrait has an MCMC function used by the authors “to test for correlated evolution between traits” and derive the above Bayes factors. Using a stepping-stone method I am unaware of. And 10⁹ iterations (repeated 3 times for checking consistency)… Reversible jump is apparently used to move between constrained and unconstrained models, leading to the pie charts at the inner nodes of the above picture. Again a by-product of BayesTrait. The trees on the left and the right are completely identical, the difference being in the inference about stratification evolution (right) and sacrifice evolution (left). While the overall hypothesis makes sense at my layman level (as a culture has to be stratified enough to impose sacrifices from its members), I am not convinced that this involved statistical analysis brings that strong a support. (But it would make a fantastic topic for an undergraduate or a Master thesis!)

approximating evidence with missing data

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , on December 23, 2015 by xi'an

University of Warwick, May 31 2010Panayiota Touloupou (Warwick), Naif Alzahrani, Peter Neal, Simon Spencer (Warwick) and Trevelyan McKinley arXived a paper yesterday on Model comparison with missing data using MCMC and importance sampling, where they proposed an importance sampling strategy based on an early MCMC run to approximate the marginal likelihood a.k.a. the evidence. Another instance of estimating a constant. It is thus similar to our Frontier paper with Jean-Michel, as well as to the recent Pima Indian survey of James and Nicolas. The authors give the difficulty to calibrate reversible jump MCMC as the starting point to their research. The importance sampler they use is the natural choice of a Gaussian or t distribution centred at some estimate of θ and with covariance matrix associated with Fisher’s information. Or derived from the warmup MCMC run. The comparison between the different approximations to the evidence are done first over longitudinal epidemiological models. Involving 11 parameters in the example processed therein. The competitors to the 9 versions of importance samplers investigated in the paper are the raw harmonic mean [rather than our HPD truncated version], Chib’s, path sampling and RJMCMC [which does not make much sense when comparing two models]. But neither bridge sampling, nor nested sampling. Without any surprise (!) harmonic means do not converge to the right value, but more surprisingly Chib’s method happens to be less accurate than most importance solutions studied therein. It may be due to the fact that Chib’s approximation requires three MCMC runs and hence is quite costly. The fact that the mixture (or defensive) importance sampling [with 5% weight on the prior] did best begs for a comparison with bridge sampling, no? The difficulty with such study is obviously that the results only apply in the setting of the simulation, hence that e.g. another mixture importance sampler or Chib’s solution would behave differently in another model. In particular, it is hard to judge of the impact of the dimensions of the parameter and of the missing data.

astronomical evidence

Posted in pictures, Statistics, University life with tags , , , , , , , , , , , , on July 24, 2015 by xi'an

As I have a huge arXiv backlog and an even higher non-arXiv backlog, I cannot be certain I will find time to comment on those three recent and quite exciting postings connecting ABC with astro- and cosmo-statistics [thanks to Ewan for pointing out those to me!]: