Archive for MCMC

g-and-k [or -h] distributions

Posted in Statistics with tags , , , , , , , , , on July 17, 2017 by xi'an

Dennis Prangle released last week an R package called gk and an associated arXived paper for running inference on the g-and-k and g-and-h quantile distributions. As should be clear from an earlier review on Karian’s and Dudewicz’s book quantile distributions, I am not particularly fond of those distributions which construction seems very artificial to me, as mostly based on the production of a closed-form quantile function. But I agree they provide a neat benchmark for ABC methods, if nothing else. However, as recently pointed out in our Wasserstein paper with Espen Bernton, Pierre Jacob and Mathieu Gerber, and explained in a post of Pierre’s on Statisfaction, the pdf can be easily constructed by numerical means, hence allows for an MCMC resolution, which is also a point made by Dennis in his paper. Using the closed-form derivation of the Normal form of the distribution [i.e., applied to Φ(x)] so that numerical derivation is not necessary.

slice sampling for Dirichlet mixture process

Posted in Books, Statistics, University life with tags , , , , , , , on June 21, 2017 by xi'an

When working with my PhD student Changye in Dauphine this morning I realised that slice sampling also applies to discrete support distributions and could even be of use in such settings. That it works is (now) straightforward in that the missing variable representation behind the slice sampler also applies to densities defined with respect to a discrete measure. That this is useful transpires from the short paper of Stephen Walker (2007) where we saw this, as Stephen relies on the slice sampler to sample from the Dirichlet mixture model by eliminating the tail problem associated with this distribution. (This paper appeared in Communications in Statistics and it is through Pati & Dunson (2014) taking advantage of this trick that Changye found about its very existence. I may have known about it in an earlier life, but I had clearly forgotten everything!)

While the prior distribution (of the weights) of the Dirichlet mixture process is easy to generate via the stick breaking representation, the posterior distribution is trickier as the weights are multiplied by the values of the sampling distribution (likelihood) at the corresponding parameter values and they cannot be normalised. Introducing a uniform to replace all weights in the mixture with an indicator that the uniform is less than those weights corresponds to a (latent variable) completion [or a demarginalisation as we called this trick in Monte Carlo Statistical Methods]. As elaborated in the paper, the Gibbs steps corresponding to this completion are easy to implement, involving only a finite number of components. Meaning the allocation to a component of the mixture can be operated rather efficiently. Or not when considering that the weights in the Dirichlet mixture are not monotone, hence that a large number of them may need to be computed before picking the next index in the mixture when the uniform draw happens to be quite small.

convergence of MCMC

Posted in Statistics with tags , , , , , , , , , on June 16, 2017 by xi'an

Michael Betancourt just posted on arXiv an historical  review piece on the convergence of MCMC, with a physical perspective.

“The success of these of Markov chain Monte Carlo, however, contributed to its own demise.”

The discourse proceeds through augmented [reality!] versions of MCMC algorithms taking advantage of the shape and nature of the target distribution, like Langevin diffusions [which cannot be simulated directly and exactly at the same time] in statistics and molecular dynamics in physics. (Which reminded me of the two parallel threads at the ICMS workshop we had a few years ago.) Merging into hybrid Monte Carlo, morphing into Hamiltonian Monte Carlo under the quills of Radford Neal and David MacKay in the 1990’s. It is a short entry (and so is this post), with some background already well-known to the community, but it nonetheless provides a perspective and references rarely mentioned in statistics.

thinning a Markov chain, statistically

Posted in Books, pictures, R, Statistics with tags , , , , , , on June 13, 2017 by xi'an

Art Owen has arXived a new version of his thinning MCMC paper, where he studies how thinning or subsampling can improve computing time in MCMC chains. I remember quite well the message set by Mark Berliner and Steve MacEachern in an early 1990’s paper that subsampling was always increasing the variance of the resulting estimators. We actually have this result in our Monte Carlo Statistical Methods book. Now, there are other perspectives on this, as for instance cases when thinning can be hard-wired by simulating directly a k-step move, delaying rejection or acceptance, prefetching, or simulating directly the accepted values as in our vanilla Rao-Blackwellisation approach. Here, Art considers the case when there is a cost θ of computing a transform of the simulation [when the transition cost a unit] and when those transforms are positively correlated with correlation ρ. Somewhat unsurprisingly, when θ is large enough, thinning becomes worth implementing. But requires extra computations in evaluating the correlation ρ and the cost θ, which is rarely comparable with the cost of computing the likelihood itself, a requirement for the Metropolis-Hastings or Hamiltonian Monte Carlo step(s).  Subsampling while keeping the right target (which is a hard constraint!) should thus have a much more effective impact on computing budgets.

accelerating MCMC

Posted in Statistics with tags , , , , , , , , , , , , on May 29, 2017 by xi'an

I have recently [well, not so recently!] been asked to write a review paper on ways of accelerating MCMC algorithms for the [review] journal WIREs Computational Statistics and would welcome all suggestions towards the goal of accelerating MCMC algorithms. Besides [and including more on]

  • coupling strategies using different kernels and switching between them;
  • tempering strategies using flatter or lower dimensional targets as intermediary steps, e.g., à la Neal;
  • sequential Monte Carlo with particle systems targeting again flatter or lower dimensional targets and adapting proposals to this effect;
  • Hamiltonian MCMC, again with connections to Radford (and more generally ways of avoiding rejections);
  • adaptive MCMC, obviously;
  • Rao-Blackwellisation, just as obviously (in the sense that increasing the precision in the resulting estimates means less simulations).

Russian roulette still rolling

Posted in Statistics with tags , , , , , , , , , , , , on March 22, 2017 by xi'an

Colin Wei and Iain Murray arXived a new version of their paper on doubly-intractable distributions, which is to be presented at AISTATS. It builds upon the Russian roulette estimator of Lyne et al. (2015), which itself exploits the debiasing technique of McLeish et al. (2011) [found earlier in the physics literature as in Carter and Cashwell, 1975, according to the current paper]. Such an unbiased estimator of the inverse of the normalising constant can be used for pseudo-marginal MCMC, except that the estimator is sometimes negative and has to be so as proved by Pierre Jacob and co-authors. As I discussed in my post on the Russian roulette estimator, replacing the negative estimate with its absolute value does not seem right because a negative value indicates that the quantity is close to zero, hence replacing it with zero would sound more appropriate. Wei and Murray start from the property that, while the expectation of the importance weight is equal to the normalising constant, the expectation of the inverse of the importance weight converges to the inverse of the weight for an MCMC chain. This however sounds like an harmonic mean estimate because the property would also stand for any substitute to the importance density, as it only requires the density to integrate to one… As noted in the paper, the variance of the resulting Roulette estimator “will be high” or even infinite. Following Glynn et al. (2014), the authors build a coupled version of that solution, which key feature is to cut the higher order terms in the debiasing estimator. This does not guarantee finite variance or positivity of the estimate, though. In order to decrease the variance (assuming it is finite), backward coupling is introduced, with a Rao-Blackwellisation step using our 1996 Biometrika derivation. Which happens to be of lower cost than the standard Rao-Blackwellisation in that special case, O(N) versus O(N²), N being the stopping rule used in the debiasing estimator. Under the assumption that the inverse importance weight has finite expectation [wrt the importance density], the resulting backward-coupling Russian roulette estimator can be proven to be unbiased, as it enjoys a finite expectation. (As in the generalised harmonic mean case, the constraint imposes thinner tails on the importance function, which then hampers the convergence of the MCMC chain.) No mention is made of achieving finite variance for those estimators, which again is a serious concern due to the similarity with harmonic means…

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.