**An** interesting query on (or from) X validated: given a Bernoulli mixture where the weights are known and the probabilities are jointly drawn from a Dirichlet, what is the most efficient from running a Gibbs sampler including the latent variables to running a basic Metropolis-Hastings algorithm based on the mixture representation to running a collapsed Gibbs sampler that only samples the indicator variables… I provided a closed form expression for the collapsed target, but believe that the most efficient solution is based on the mixture representation!

## Archive for Gibbs sampler

## Bernoulli mixtures

Posted in pictures, Statistics, University life with tags Bernoulli mixture, cross validated, Gibbs sampler, Helvetia, Jakob Bernoulli, Metropolis-Hastings algorithm, mixtures, stamp on October 30, 2019 by xi'an## A precursor of ABC-Gibbs

Posted in Books, R, Statistics with tags ABC, ABC-Gibbs, compatible conditional distributions, Genetics, Gibbs sampler, high dimensions, incoherent inference, incompatible conditionals, insufficiency, likelihood-free methods, sufficient statistics on June 7, 2019 by xi'an**F**ollowing our arXival of ABC-Gibbs, Dennis Prangle pointed out to us a 2016 paper by Athanasios Kousathanas, Christoph Leuenberger, Jonas Helfer, Mathieu Quinodoz, Matthieu Foll, and Daniel Wegmann, Likelihood-Free Inference in High-Dimensional Model, published in Genetics, Vol. 203, 893–904 in June 2016. This paper contains a version of ABC Gibbs where parameters are sequentially simulated from conditionals that depend on the data only through small dimension conditionally sufficient statistics. I had actually blogged about this paper in 2015 but since then completely forgotten about it. (The comments I had made at the time still hold, already pertaining to the coherence or lack thereof of the sampler. I had also forgotten I had run an experiment of an exact Gibbs sampler with incoherent conditionals, which then seemed to converge to something, if not the exact posterior.)

All ABC algorithms, including ABC-PaSS introduced here, require that statistics are sufficient for estimating the parameters of a given model. As mentioned above, parameter-wise sufficient statistics as required by ABC-PaSS are trivial to find for distributions of the exponential family. Since many population genetics models do not follow such distributions, sufficient statistics are known for the most simple models only. For more realistic models involving multiple populations or population size changes, only approximately-sufficient statistics can be found.

While Gibbs sampling is not mentioned in the paper, this is indeed a form of ABC-Gibbs, with the advantage of not facing convergence issues thanks to the sufficiency. The drawback being that this setting is restricted to exponential families and hence difficult to extrapolate to non-exponential distributions, as using almost-sufficient (or not) summary statistics leads to incompatible conditionals and thus jeopardise the convergence of the sampler. When thinking a wee bit more about the case treated by Kousathanas et al., I am actually uncertain about the validation of the sampler. When tolerance is equal to zero, this is not an issue as it reproduces the regular Gibbs sampler. Otherwise, each conditional ABC step amounts to introducing an auxiliary variable represented by the simulated summary statistic. Since the distribution of this summary statistic depends on more than the parameter for which it is sufficient, in general, it should also appear in the conditional distribution of other parameters. At least from this Gibbs perspective, it thus relies on incompatible conditionals, which makes the conditions proposed in our own paper the more relevant.

## ABC with Gibbs steps

Posted in Statistics with tags ABC, ABC-Gibbs, Approximate Bayesian computation, Bayesian inference, bois de Boulogne, compatible conditional distributions, contraction, convergence, ergodicity, France, Gibbs sampler, hierarchical Bayesian modelling, incompatible conditionals, La Défense, Paris, stationarity, tolerance, Université Paris Dauphine on June 3, 2019 by xi'an**W**ith Grégoire Clarté, Robin Ryder and Julien Stoehr, all from Paris-Dauphine, we have just arXived a paper on the specifics of ABC-Gibbs, which is a version of ABC where the generic ABC accept-reject step is replaced by a sequence of n conditional ABC accept-reject steps, each aiming at an ABC version of a conditional distribution extracted from the joint and intractable target. Hence an ABC version of the standard Gibbs sampler. What makes it so special is that each conditional can (and should) be conditioning on a different statistic in order to decrease the dimension of this statistic, ideally down to the dimension of the corresponding component of the parameter. This successfully bypasses the curse of dimensionality but immediately meets with two difficulties. The first one is that the resulting sequence of conditionals is not coherent, since it is not a Gibbs sampler on the ABC target. The conditionals are thus incompatible and therefore convergence of the associated Markov chain becomes an issue. We produce sufficient conditions for the Gibbs sampler to converge to a stationary distribution using incompatible conditionals. The second problem is then that, provided it exists, the limiting and also intractable distribution does not enjoy a Bayesian interpretation, hence may fail to be justified from an inferential viewpoint. We however succeed in producing a version of ABC-Gibbs in a hierarchical model where the limiting distribution can be explicited and even better can be weighted towards recovering the original target. (At least with limiting zero tolerance.)

## Roberto Casarin’s talk at CREST tomorrow

Posted in Statistics with tags Bayesian econometrics, Ca' Foscari University, CREST, Data augmentation, financial network, Gibbs sampler, Gran Canale, hidden Markov chain, Markov switching models, seminar, Université Paris-Saclay, Venezia on March 13, 2019 by xi'an**M**y former student and friend Roberto Casarin (University Ca’Foscari, Venice) will talk tomorrow at the CREST Financial Econometrics seminar on

“Bayesian Markov Switching Tensor Regression for Time-varying Networks”

Time: 10:30

Date: 14 March 2019

Place: Room 3001, ENSAE, Université Paris-Saclay

Abstract : We propose a new Bayesian Markov switching regression model for multi-dimensional arrays (tensors) of binary time series. We assume a zero-inflated logit dynamics with time-varying parameters and apply it to multi-layer temporal networks. The original contribution is threefold. First, in order to avoid over-fitting we propose a parsimonious parameterisation of the model, based on a low-rank decomposition of the tensor of regression coefficients. Second, the parameters of the tensor model are driven by a hidden Markov chain, thus allowing for structural changes. The regimes are identified through prior constraints on the mixing probability of the zero-inflated model. Finally, we model the jointly dynamics of the network and of a set of variables of interest. We follow a Bayesian approach to inference, exploiting the Pólya-Gamma data augmentation scheme for logit models in order to provide an efficient Gibbs sampler for posterior approximation. We show the effectiveness of the sampler on simulated datasets of medium-big sizes, finally we apply the methodology to a real dataset of financial networks.

## more multiple proposal MCMC

Posted in Books, Statistics with tags delayed rejection sampling, directed acyclic graphs, Gibbs sampler, multiple-try Metropolis algorithm, parallelisation, prefetching, pseudo-posterior, subsampling on July 26, 2018 by xi'an**L**uo and Tjelmeland just arXived a paper on a new version of multiple-try Metropolis Hastings, the addendum being in defining the additional proposed copies via a dependence graph like (a) above, with one version from the target and all others from operational and conditional proposal kernels. Respecting the dependence graph, as in (b). As I did, you may then wonder where both the graph and the conditional do come from. Which reminds me of the pseudo-posteriors of Carlin and Chib (1995), even though this is not terribly connected. Green (1995).) (But not disconnected either since the authors mention But, given the graph, following a Gibbs scheme, one of the 17 nodes is chosen as generated from the target, using the proper conditional on that index [which is purely artificial from the point of view of the original simulation problem!]. As noted above, the graph is an issue, but since it is artificial, it can be devised to either allow for quasi-independence between the proposed values or on the opposite to induce long range dependence, which corresponds to conducting multiple MCMC steps before reaching the end nodes, a feature that is very appealing in my opinion. And reminds me of prefetching. (As I am listening to Nicolas Chopin’s lecture in Warwick at the moment, there also seems to be a connection with pMCMC.) Still, I remain unclear as to the devising of the graph of dependent proposals, as its depth should be somehow connected with the mixing properties of the original proposal. Gains in convergence may thus come at a high cost at the construction stage.

## another version of the corrected harmonic mean estimator

Posted in Books, pictures, Statistics, University life with tags Gibbs sampler, harmonic mean estimator, HPD region, importance sampling, MCMC algorithm, Monte Carlo Statistical Methods on June 11, 2018 by xi'an**A** few days ago I came across a short paper in the Central European Journal of Economic Modelling and Econometrics by Pajor and Osiewalski that proposes a correction to the infamous harmonic mean estimator that is essentially the one Darren and I made in 2009, namely to restrict the evaluations of the likelihood function to a subset **A** of the simulations from the posterior. Paper that relates to an earlier 2009 paper by Peter Lenk, which investigates the same object with this same proposal and that we had missed for all that time. The difference is that, while we examine an arbitrary HPD region at level 50% or 80% as the subset **A**, Lenk proposes to derive a minimum likelihood value from the MCMC run and to use the associated HPD region, which means using all simulations, hence producing the same object as the original harmonic mean estimator, except that it is corrected by a multiplicative factor P(**A**). Or rather an approximation. This correction thus maintains the infinite variance of the original, a point apparently missed in the paper.

## amazing appendix

Posted in Books, Statistics, Travel, University life with tags auxiliary variable, Colorado, Fort Collins, Gibbs sampler, Julian Besag, MCMC, Metropolis-within-Gibbs algorithm, Monte Carlo Statistical Methods, Oxford, random simulation, simulation, Statistical Science on February 13, 2018 by xi'an**I**n the first appendix of the 1995 Statistical Science paper of Besag, Green, Higdon and Mengersen, on MCMC, “Bayesian Computation and Stochastic Systems”, stands a fairly neat result I was not aware of (and which Arnaud Doucet, with his unrivalled knowledge of the literature!, pointed out to me in Oxford, avoiding me the tedium to try to prove it afresco!). I remember well reading a version of the paper in Fort Collins, Colorado, in 1993 (I think!) but nothing about this result.

It goes as follows: when running a Metropolis-within-Gibbs sampler for component x¹ of a collection of variates x¹,x²,…, thus aiming at simulating from the full conditional of x¹ given x⁻¹ by making a proposal q(x|x¹,x⁻¹), it is perfectly acceptable to use a proposal that depends on a parameter α (no surprise so far!) *and* to generate this parameter α anew at each iteration (still unsurprising as α can be taken as an auxiliary variable) *and* to have the distribution of this parameter α depending on the other variates x²,…, i.e., x⁻¹. This is the surprising part, as adding α as an auxiliary variable was messing up the update of x⁻¹. But the proof as found in the 1995 paper [page 35] does not require to consider α as such as it establishes global balance directly. (Or maybe still detailed balance when writing the whole Gibbs sampler as a cycle of Metropolis steps.) Terrific! And a whiff mysterious..!