Archive for MCMC algorithms

Bayesian inference with intractable normalizing functions

Posted in Books, Statistics with tags , , , , , , , , , , , on December 13, 2018 by xi'an

In the latest September issue of JASA I received a few days ago, I spotted a review paper by Jaewoo Park & Murali Haran on intractable normalising constants Z(θ). There have been many proposals for solving this problem as well as several surveys, some conferences and even a book. The current survey focus on MCMC solutions, from auxiliary variable approaches to likelihood approximation algorithms (albeit without ABC entries, even though the 2006 auxiliary variable solutions of Møller et al. et of Murray et al. do simulate pseudo-observations and hence…). This includes the MCMC approximations to auxiliary sampling proposed by Faming Liang and co-authors across several papers. And the paper Yves Atchadé, Nicolas Lartillot and I wrote ten years ago on an adaptive MCMC targeting Z(θ) and using stochastic approximation à la Wang-Landau. Park & Haran stress the relevance of using sufficient statistics in this approach towards fighting computational costs, which makes me wonder if an ABC version could be envisioned.  The paper also includes pseudo-marginal techniques like Russian Roulette (once spelled Roullette) and noisy MCMC as proposed in Alquier et al.  (2016). These methods are compared on three examples: (1) the Ising model, (2) a social network model, the Florentine business dataset used in our original paper, and a larger one where most methods prove too costly, and (3) an attraction-repulsion point process model. In conclusion, an interesting survey, taking care to spell out the calibration requirements and the theoretical validation, if of course depending on the chosen benchmarks.

Gibbs for incompatible kids

Posted in Books, Statistics, University life with tags , , , , , , , , , , on September 27, 2018 by xi'an

In continuation of my earlier post on Bayesian GANs, which resort to strongly incompatible conditionals, I read a 2015 paper of Chen and Ip that I had missed. (Published in the Journal of Statistical Computation and Simulation which I first confused with JCGS and which I do not know at all. Actually, when looking at its editorial board,  I recognised only one name.) But the study therein is quite disappointing and not helping as it considers Markov chains on finite state spaces, meaning that the transition distributions are matrices, meaning also that convergence is ensured if these matrices have no null probability term. And while the paper is motivated by realistic situations where incompatible conditionals can reasonably appear, the paper only produces illustrations on two and three states Markov chains. Not that helpful, in the end… The game is still afoot!

subset sampling

Posted in Statistics with tags , , , , , , , , on July 13, 2018 by xi'an

A paper by Au and Beck (2001) was mentioned during a talk at MCqMC 2018 in Rennes and I checked Probabilistic Engineering Mechanics for details. There is no clear indication that the subset simulation advocated therein is particularly effective. The core idea is to obtain the probability to belong to a small set A by a cascading formula, namely the product of the probability to belong to A¹, then the conditional probability to belong to A² given A¹, &tc. When the subsets A¹, A², …, A constitute a decreasing embedded sequence. The simulation conditional on being in one of the subsets A^i is operated by a random-walk Metropolis-within-Gibbs scheme, with an additional rejection when the value is not in the said subset. (Surprisingly, the authors re-establish the validity of this scheme.) Hence the proposal faces similar issues as nested sampling, except that the nested subsets here are defined quite differently as they are essentially free, provided they can be easily evaluated. Each of the random walks need be scaled, the harder a task because this depends on the corresponding subset volume. The subsets A^i themselves are rarely defined in a natural manner, except when being tail events. And need to be calibrated so that the conditional probability of falling into each remains large enough, the cost of free choice. The Markov chain on the previous subset A^i can prove useful to build the next subset A^{i+1}, but there is no general principle behind this remark. (If any, this is connected with X entropy.) But else, the past chains are very much wasted, compared with, say, an SMC treatment of the problem. The paper also notices that starting a Markov chain in the set A^{i+1} means there is no burnin time and hence that the probability estimators are thus unbiased. (This creates a correlation between successive Markov chains, but I think it could be ignored if the starting point was chosen at random or after a random number of extra steps.) The authors further point out that the chain may fail to be ergodic, if the proposal distribution lacks energy to link connected regions of the current subset A^i. They suggest using multiple chains with multiple starting points, which alleviates the issue only to some extent, as it ultimately depends on the spread of the starting points. As acknowledged in the paper.

MCqMC 2018, Rennes [slides]

Posted in Statistics with tags , , , , , on July 3, 2018 by xi'an

Here are my slides for the talk I give this morning at MCqMC 20188. Based on slides first written by Changye Wu and on our joint papers. As it happens, I was under the impression I would give a survey on partially deterministic Markov processes. But, as it goes (!), my talk takes place after a superb plenary talk by Christophe Andrieu on non-reversibility, where he gave motivations for recoursing to non-reversibility and general results for variance reduction, plus a whole session on the topic by Jorens Bierkens, Alex Thiéry, Alain Durmus, and Arnak Dalalyan (CREST), which covered the topics in the following slides, only better! Reducing the informative contents of my talk to the alternative to the Zig-Zag sampler Changye proposed, which makes the talk of limited appeal, I am afraid. (There are four other sessions at the same time, fortunately!)

Metropolis-Hastings importance sampling

Posted in Books, Statistics, University life with tags , , , , , , , , , on June 6, 2018 by xi'an

[Warning: As I first got the paper from the authors and sent them my comments, this paper read contains their reply as well.]

In a sort of crazy coincidence, Daniel Rudolf and Björn Sprungk arXived a paper on a Metropolis-Hastings importance sampling estimator that offers similarities with  the one by Ingmar Schuster and Ilja Klebanov posted on arXiv the same day. The major difference in the construction of the importance sampler is that Rudolf and Sprungk use the conditional distribution of the proposal in the denominator of their importance weight, while Schuster and Klebanov go for the marginal (or a Rao-Blackwell representation of the marginal), mostly in an independent Metropolis-Hastings setting (for convergence) and for a discretised Langevin version in the applications. The former use a very functional L² approach to convergence (which reminded me of the early Schervish and Carlin, 1990, paper on the convergence of MCMC algorithms), not all of it necessary in my opinion. As for instance the extension of convergence properties to the augmented chain, namely (current, proposed), is rather straightforward since the proposed chain is a random transform of the current chain. An interesting remark at the end of the proof of the CLT is that the asymptotic variance of the importance sampling estimator is the same as with iid realisations from the target. This is a point we also noticed when constructing population Monte Carlo techniques (more than ten years ago), namely that dependence on the past in sequential Monte Carlo does not impact the validation and the moments of the resulting estimators, simply because “everything cancels” in importance ratios. The mean square error bound on the Monte Carlo error (Theorem 20) is not very surprising as the term ρ(y)²/P(x,y) appears naturally in the variance of importance samplers.

The first illustration where the importance sampler does worse than the initial MCMC estimator for a wide range of acceptance probabilities (Figures 2 and 3, which is which?) and I do not understand the opposite conclusion from the authors.

[Here is an answer from Daniel and Björn about this point:]

Indeed the formulation in our paper is unfortunate. The point we want to stress is that we observed in the numerical experiments certain ranges of step-sizes for which MH importance sampling shows a better performance than the classical MH algorithm with optimal scaling. Meaning that the MH importance sampling with optimal step-size can outperform MH sampling, without using additional computational resources. Surprisingly, the optimal step-size for the MH importance sampling estimator seems to remain constant for an increasing dimension in contrast to the well-known optimal scaling of the MH algorithm (given by a constant optimal acceptance rate).

The second uses the Pima Indian diabetes benchmark, amusingly (?) referring to Chopin and Ridgway (2017) who warn against the recourse to this dataset and to this model! The loss in mean square error due to the importance sampling may again be massive (Figure 5) and setting for an optimisation of the scaling factor in Metropolis-Hastings algorithms sounds unrealistic.

[And another answer from Daniel and Björn about this point:]

Indeed, Chopin and Ridgway suggest more complex problems with a larger number of covariates as benchmarks. However, the well-studied PIMA data set is a sufficient example in order to illustrate the possible benefits but also the limitations of the MH importance sampling approach. The latter are clearly (a) the required knowledge about the optimal step-size—otherwise the performance can indeed be dramatically worse than for the MH algorithm—and (b) the restriction to a small or at most moderate number of covariates. As you are indicating, optimizing the scaling factor is a challenging task. However, the hope is to derive some simple rule of thumb for the MH importance sampler similar to the well-known acceptance rate tuning for the standard MCMC estimator.

Markov chain importance sampling

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , on May 31, 2018 by xi'an

Ingmar Schuster (formerly a postdoc at Dauphine and now in Freie Universität Berlin) and Ilja Klebanov (from Berlin) have recently arXived a paper on recycling proposed values in [a rather large class of] Metropolis-Hastings and unadjusted Langevin algorithms. This means using the proposed variates of one of these algorithms as in an importance sampler, with an importance weight going from the target over the (fully conditional) proposal to the target over the marginal stationary target. In the Metropolis-Hastings case, since the later is not available in most setups, the authors suggest using a Rao-Blackwellised nonparametric estimate based on the entire MCMC chain. Or a subset.

“Our estimator refutes the folk theorem that it is hard to estimate [the normalising constant] with mainstream Monte Carlo methods such as Metropolis-Hastings.”

The paper thus brings an interesting focus on the proposed values, rather than on the original Markov chain,  which naturally brings back to mind the derivation of the joint distribution of these proposed values we made in our (1996) Rao-Blackwellisation paper with George Casella. Where we considered a parametric and non-asymptotic version of this distribution, which brings a guaranteed improvement to MCMC (Metropolis-Hastings) estimates of integrals. In subsequent papers with George, we tried to quantify this improvement and to compare different importance samplers based on some importance sampling corrections, but as far as I remember, we only got partial results along this way, and did not cover the special case of the normalising constant Þ… Normalising constants did not seem such a pressing issue at that time, I figure. (A Monte Carlo 101 question: how can we be certain the importance sampler offers a finite variance?)

Ingmar’s views about this:

I think this is interesting future work. My intuition is that for Metropolis-Hastings importance sampling with random walk proposals, the variance is guaranteed to be finite because the importance distribution ρ_θ is a convolution of your target ρ with the random walk kernel q. This guarantees that the tails of ρ_θ are no lighter than those of ρ. What other forms of q mean for the tails of ρ_θ I have less intuition about.

When considering the Langevin alternative with transition (4), I was first confused and thought it was incorrect for moving from one value of Y (proposal) to the next. But that’s what unadjusted means in “unadjusted Langevin”! As pointed out in the early Langevin literature, e.g., by Gareth Roberts and Richard Tweedie, using a discretised Langevin diffusion in an MCMC framework means there is a risk of non-stationarity & non-ergodicity. Obviously, the corrected (MALA) version is more delicate to approximate (?) but at the very least it ensures the Markov chain does not diverge. Even when the unadjusted Langevin has a stationary regime, its joint distribution is likely quite far from the joint distribution of a proper discretisation. Now this also made me think about a parameterised version in the 1996 paper spirit, but there is nothing specific about MALA that would prevent the implementation of the general principle. As for the unadjusted version, the joint distribution is directly available.  (But not necessarily the marginals.)

Here is an answer from Ingmar about that point

Personally, I think the most interesting part is the practical performance gain in terms of estimation accuracy for fixed CPU time, combined with the convergence guarantee from the CLT. ULA was particularly important to us because of the papers of Arnak Dalalyan, Alain Durmus & Eric Moulines and recently from Mike Jordan’s group, which all look at an unadjusted Langevin diffusion (and unimodal target distributions). But MALA admits a Metropolis-Hastings importance sampling estimator, just as Random Walk Metropolis does – we didn’t include MALA in the experiments to not get people confused with MALA and ULA. But there is no delicacy involved whatsoever in approximating the marginal MALA proposal distribution. The beauty of our approach is that it works for almost all Metropolis-Hastings algorithms where you can evaluate the proposal density q, there is no constraint to use random walks at all (we will emphasize this more in the paper).

adaptive independent Metropolis-Hastings

Posted in Statistics with tags , , , , , , on May 8, 2018 by xi'an

When rereading this paper by Halden et al. (2009), I was reminded of the earlier and somewhat under-appreciated Gåsemyr (2003). But I find the convergence results therein rather counter-intuitive in that they seem to justify adaptive independent proposals with no strong requirement. Besides the massive Doeblin condition:

“The Doeblin condition essentially requires that all the proposal distribution [sic] has uniformly heavier tails than the target distribution.”

Even when the adaptation is based on an history vector made of rejected values and non-replicated accepted values. Actually  convergence of this sequence of adaptive proposals kernels is established under a concentration of the Doeblin constants a¹,a²,… towards one, in the sense that

E[(1-a¹)(1-a²)…]=0.

The reason may be that, with chains satisfying a Doeblin condition, there is a probability to reach stationarity at each step. Equal to a¹, a², … And hence to ignore adaptivity since each kernel keep the target π invariant. So in the end this is not so astounding. (The paper also reminded me of Wolfgang [or Vincent] Doeblin‘s short and tragic life.)