Archive for Rao-Blackwellisation

Quasi-Monte Carlo sampling

Posted in Books, Kids, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , on December 10, 2014 by xi'an

RSS wine“The QMC algorithm forces us to write any simulation as an explicit function of uniform samples.” (p.8)

As posted a few days ago, Mathieu Gerber and Nicolas Chopin will read this afternoon a Paper to the Royal Statistical Society on their sequential quasi-Monte Carlo sampling paper.  Here are some comments on the paper that are preliminaries to my written discussion (to be sent before the slightly awkward deadline of Jan 2, 2015).

Quasi-Monte Carlo methods are definitely not popular within the (mainstream) statistical community, despite regular attempts by respected researchers like Art Owen and Pierre L’Écuyer to induce more use of those methods. It is thus to be hoped that the current attempt will be more successful, it being Read to the Royal Statistical Society being a major step towards a wide diffusion. I am looking forward to the collection of discussions that will result from the incoming afternoon (and bemoan once again having to miss it!).

“It is also the resampling step that makes the introduction of QMC into SMC sampling non-trivial.” (p.3)

At a mathematical level, the fact that randomised low discrepancy sequences produce both unbiased estimators and error rates of order

\mathfrak{O}(N^{-1}\log(N)^{d-}) \text{ at cost } \mathfrak{O}(N\log(N))

means that randomised quasi-Monte Carlo methods should always be used, instead of regular Monte Carlo methods! So why is it not always used?! The difficulty stands [I think] in expressing the Monte Carlo estimators in terms of a deterministic function of a fixed number of uniforms (and possibly of past simulated values). At least this is why I never attempted at crossing the Rubicon into the quasi-Monte Carlo realm… And maybe also why the step had to appear in connection with particle filters, which can be seen as dynamic importance sampling methods and hence enjoy a local iid-ness that relates better to quasi-Monte Carlo integrators than single-chain MCMC algorithms.  For instance, each resampling step in a particle filter consists in a repeated multinomial generation, hence should have been turned into quasi-Monte Carlo ages ago. (However, rather than the basic solution drafted in Table 2, lower variance solutions like systematic and residual sampling have been proposed in the particle literature and I wonder if any of these is a special form of quasi-Monte Carlo.) In the present setting, the authors move further and apply quasi-Monte Carlo to the particles themselves. However, they still assume the deterministic transform

\mathbf{x}_t^n = \Gamma_t(\mathbf{x}_{t-1}^n,\mathbf{u}_{t}^n)

which the q-block on which I stumbled each time I contemplated quasi-Monte Carlo… So the fundamental difficulty with the whole proposal is that the generation from the Markov proposal

m_t(\tilde{\mathbf{x}}_{t-1}^n,\cdot)

has to be of the above form. Is the strength of this assumption discussed anywhere in the paper? All baseline distributions there are normal. And in the case it does not easily apply, what would the gain bw in only using the second step (i.e., quasi-Monte Carlo-ing the multinomial simulation from the empirical cdf)? In a sequential setting with unknown parameters θ, the transform is modified each time θ is modified and I wonder at the impact on computing cost if the inverse cdf is not available analytically. And I presume simulating the θ’s cannot benefit from quasi-Monte Carlo improvements.

The paper obviously cannot get into every detail, obviously, but I would also welcome indications on the cost of deriving the Hilbert curve, in particular in connection with the dimension d as it has to separate all of the N particles, and on the stopping rule on m that means only Hm is used.

Another question stands with the multiplicity of low discrepancy sequences and their impact on the overall convergence. If Art Owen’s (1997) nested scrambling leads to the best rate, as implied by Theorem 7, why should we ever consider another choice?

In connection with Lemma 1 and the sequential quasi-Monte Carlo approximation of the evidence, I wonder at any possible Rao-Blackwellisation using all proposed moves rather than only those accepted. I mean, from a quasi-Monte Carlo viewpoint, is Rao-Blackwellisation easier and is it of any significant interest?

What are the computing costs and gains for forward and backward sampling? They are not discussed there. I also fail to understand the trick at the end of 4.2.1, using SQMC on a single vector instead of (t+1) of them. Again assuming inverse cdfs are available? Any connection with the Polson et al.’s particle learning literature?

Last questions: what is the (learning) effort for lazy me to move to SQMC? Any hope of stepping outside particle filtering?

importance sampling schemes for evidence approximation [revised]

Posted in Statistics, University life with tags , , , , , , , on November 18, 2014 by xi'an

After a rather intense period of new simulations and versions, Juong Een (Kate) Lee and I have now resubmitted our paper on (some) importance sampling schemes for evidence approximation in mixture models to Bayesian Analysis. There is no fundamental change in the new version but rather a more detailed description of what those importance schemes mean in practice. The original idea in the paper is to improve upon the Rao-Blackwellisation solution proposed by Berkoff et al. (2002) and later by Marin et al. (2005) to avoid the impact of label switching on Chib’s formula. The Rao-Blackwellisation consists in averaging over all permutations of the labels while the improvement relies on the elimination of useless permutations, namely those that produce a negligible conditional density in Chib’s (candidate’s) formula. While the improvement implies truncated the overall sum and hence induces a potential bias (which was the concern of one referee), the determination of the irrelevant permutations after relabelling next to a single mode does not appear to cause any bias, while reducing the computational overload. Referees also made us aware of many recent proposals that conduct to different evidence approximations, albeit not directly related with our purpose. (One was Rodrigues and Walker, 2014, discussed and commented in a recent post.)

density normalization for MCMC algorithms

Posted in Statistics, University life with tags , , , , , , , , on November 6, 2014 by xi'an

Another paper addressing the estimation of the normalising constant and the wealth of available solutions just came out on arXiv, with the full title of “Target density normalization for Markov chain Monte Carlo algorithms“, written by Allen Caldwell and Chang Liu. (I became aware of it by courtesy of Ewan Cameron, as it appeared in the physics section of arXiv. It is actually a wee bit annoying that papers in the subcategory “Data Analysis, Statistics and Probability” of physics do not get an automated reposting on the statistics lists…)

In this paper, the authors compare three approaches to the problem of finding

\mathfrak{I} = \int_\Omega f(\lambda)\,\text{d}\lambda

when the density f is unormalised, i.e., in more formal terms, when f is proportional to a probability density (and available):

  1. an “arithmetic mean”, which is an importance sampler based on (a) reducing the integration volume to a neighbourhood ω of the global mode. This neighbourhood is chosen as an hypercube and the importance function turns out to be the uniform over this hypercube. The corresponding estimator is then a rescaled version of the average of f over uniform simulations in ω.
  2.  an “harmonic mean”, of all choices!, with again an integration over the neighbourhood ω of the global mode in order to avoid the almost sure infinite variance of harmonic mean estimators.
  3. a Laplace approximation, using the target at the mode and the Hessian at the mode as well.

The paper then goes to comparing those three solutions on a few examples, demonstrating how the diameter of the hypercube can be calibrated towards a minimum (estimated) uncertainty. The rather anticlimactic conclusion is that the arithmetic mean is the most reliable solution as harmonic means may fail in larger dimension and more importantly fail to signal its failure, while Laplace approximations only approximate well quasi-Gaussian densities…

What I find most interesting in this paper is the idea of using only one part of the integration space to compute the integral, even though it is not exactly new. Focussing on a specific region ω has pros and cons, the pros being that the reduction to a modal region reduces needs for absolute MCMC convergence and helps in selecting alternative proposals and also prevents from the worst consequences of using a dreaded harmonic mean, the cons being that the region needs be well-identified, which means requirements on the MCMC kernel, and that the estimate is a product of two estimates, the frequency being driven by a Binomial noise.  I also like very much the idea of calibrating the diameter Δof the hypercube ex-post by estimating the uncertainty.

As an aside, the paper mentions most of the alternative solutions I just presented in my Monte Carlo graduate course two days ago (like nested or bridge or Rao-Blackwellised sampling, including our proposal with Darren Wraith), but dismisses them as not “directly applicable in an MCMC setting”, i.e., without modifying this setting. I unsurprisingly dispute this labelling, both because something like the Laplace approximation requires extra-work on the MCMC output (and once done this work can lead to advanced Laplace methods like INLA) and because other methods could be considered as well (for instance, bridge sampling over several hypercubes). As shown in the recent paper by Mathieu Gerber and Nicolas Chopin (soon to be discussed at the RSS!), MCqMC has also become a feasible alternative that would compete well with the methods studied in this paper.

Overall, this is a paper that comes in a long list of papers on constant approximations. I do not find the Markov chain of MCMC aspect particularly compelling or specific, once the effective sample size is accounted for. It would be nice to find generic ways of optimising the visit to the hypercube ω and to estimate efficiently the weight of ω. The comparison is solely run over examples, but they all rely on a proper characterisation of the hypercube and the ability to simulate efficiently f over that hypercube.

Combining Particle MCMC with Rao-Blackwellized Monte Carlo Data Association

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on October 10, 2014 by xi'an

This recently arXived paper by Juho Kokkala and Simo Särkkä mixes a whole lot of interesting topics, from particle MCMC and Rao-Blackwellisation to particle filters, Kalman filters, and even bear population estimation. The starting setup is the state-space hidden process models where particle filters are of use. And where Andrieu, Doucet and Hollenstein (2010) introduced their particle MCMC algorithms. Rao-Blackwellisation steps have been proposed in this setup in the original paper, as well as in the ensuing discussion, like recycling rejected parameters and associated particles. The beginning of the paper is a review of the literature in this area, in particular of the Rao-Blackwellized Monte Carlo Data Association algorithm developed by Särkkä et al. (2007), of which I was not aware previously. (I alas have not followed closely enough the filtering literature in the past years.) Targets evolve independently according to Gaussian dynamics.

In the description of the model (Section 3), I feel there are prerequisites on the model I did not have (and did not check in Särkkä et al., 2007), like the meaning of targets and measurements: it seems the model assumes each measurement corresponds to a given target. More details or an example would have helped. The extension against the existing appears to be the (major) step of including unknown parameters. Due to my lack of expertise in the domain, I have no notion of the existence of similar proposals in the literature, but handling unknown parameters is definitely of direct relevance for the statistical analysis of such problems!

The simulation experiment based on an Ornstein-Uhlenbeck model is somewhat anticlimactic in that the posterior on the mean reversion rate is essentially the prior, conveniently centred at the true value, while the others remain quite wide. It may be that the experiment was too ambitious in selecting 30 simultaneous targets with only a total of 150 observations. Without highly informative priors, my beotian reaction is to doubt the feasibility of the inference. In the case of the Finnish bear study, the huge discrepancy between priors and posteriors, as well as the significant difference between the forestry expert estimations and the model predictions should be discussed, if not addressed, possibly via a simulation using the posteriors as priors. Or maybe using a hierarchical Bayes model to gather a time-wise coherence in the number of bear families. (I wonder if this technique would apply to the type of data gathered by Mohan Delampady on the West Ghats tigers…)

Overall, I am slightly intrigued by the practice of running MCMC chains in parallel and merging the outcomes with no further processing. This assumes a lot in terms of convergence and mixing on all the chains. However, convergence is never directly addressed in the paper.

efficient exploration of multi-modal posterior distributions

Posted in Books, Statistics, University life with tags , , , , on September 1, 2014 by xi'an

The 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.

recycling accept-reject rejections

Posted in Statistics, University life with tags , , , , , , , , , on July 1, 2014 by xi'an

Vinayak 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

p(x|\theta) \propto f(x|\theta) \le q(x|\theta) M\,,

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

\prod_{i=1}^n \dfrac{f(x_i|\theta)}{M} \prod_{j=1}^{m_i} \left\{q(y_{ij}|\theta) -\dfrac{f(y_{ij}|\theta)}{M}\right\}

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

Now 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”.)

It 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

  1. in closed form;
  2. with a manageable and tight enough “constant” M;
  3. 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.

The 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).

Note 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…

[more] parallel MCMC

Posted in Books, Mountains with tags , , , , , , , , , , on April 3, 2014 by xi'an

Scott Schmidler and his Ph.D. student Douglas VanDerwerken have arXived a paper on parallel MCMC the very day I left for Chamonix, prior to MCMSki IV, so it is no wonder I missed it at the time. This work is somewhat in the spirit of the parallel papers Scott et al.’s consensus Bayes,  Neiswanger et al.’s embarrassingly parallel MCMC, Wang and Dunson’s Weierstrassed MCMC (and even White et al.’s parallel ABC), namely that the computation of the likelihood can be broken into batches and MCMC run over those batches independently. In their short survey of previous works on parallelization, VanDerwerken and Schmidler overlooked our neat (!) JCGS Rao-Blackwellisation with Pierre Jacob and Murray Smith, maybe because it sounds more like post-processing than genuine parallelization (in that it does not speed up the convergence of the chain but rather improves the Monte Carlo usages one can make of this chain), maybe because they did not know of it.

“This approach has two shortcomings: first, it requires a number of independent simulations, and thus processors, equal to the size of the partition; this may grow exponentially in dim(Θ). Second, the rejection often needed for the restriction doesn’t permit easy evaluation of transition kernel densities, required below. In addition, estimating the relative weights wi with which they should be combined requires care.” (p.3)

The idea of the authors is to replace an exploration of the whole space operated via a single Markov chain (or by parallel chains acting independently which all have to “converge”) with parallel and independent explorations of parts of the space by separate Markov chains. “Small is beautiful”: it takes a shorter while to explore each set of the partition, hence to converge, and, more importantly, each chain can work in parallel to the others. More specifically, given a partition of the space, into sets Ai with posterior weights wi, parallel chains are associated with targets equal to the original target restricted to those Ai‘s. This is therefore an MCMC version of partitioned sampling. With regard to the shortcomings listed in the quote above, the authors consider that there does not need to be a bijection between the partition sets and the chains, in that a chain can move across partitions and thus contribute to several integral evaluations simultaneously. I am a bit worried about this argument since it amounts to getting a random number of simulations within each partition set Ai. In my (maybe biased) perception of partitioned sampling, this sounds somewhat counter-productive, as it increases the variance of the overall estimator. (Of course, not restricting a chain to a given partition set Ai has the incentive of avoiding a possibly massive amount of rejection steps. It is however unclear (a) whether or not it impacts ergodicity (it all depends on the way the chain is constructed, i.e. against which target(s)…) as it could lead to an over-representation of some boundaries and (b) whether or not it improves the overall convergence properties of the chain(s).)

“The approach presented here represents a solution to this problem which can completely remove the waiting times for crossing between modes, leaving only the relatively short within-mode equilibration times.” (p.4)

A more delicate issue with the partitioned MCMC approach (in my opinion!) stands with the partitioning. Indeed, in a complex and high-dimension model, the construction of the appropriate partition is a challenge in itself as we often have no prior idea where the modal areas are. Waiting for a correct exploration of the modes is indeed faster than waiting for crossing between modes, provided all modes are represented and the chain for each partition set Ai has enough energy to explore this set. It actually sounds (slightly?) unlikely that a target with huge gaps between modes will see a considerable improvement from the partioned version when the partition sets Ai are selected on the go, because some of the boundaries between the partition sets may be hard to reach with a off-the-shelf proposal. (Obviously, the second part of the method on the adaptive construction of partitions is yet in the writing and I am looking forward its aXival!)

Furthermore, as noted by Pierre Jacob (of Statisfaction fame!), the adaptive construction of the partition has a lot in common with Wang-Landau schemes. Which goal is to produce a flat histogram proposal from the current exploration of the state space. Connections with Atchadé’s and Liu’s (2010, Statistical Sinica) extension of the original Wang-Landau algorithm could have been spelled out. Esp. as the Voronoï tessellation construct seems quite innovative in this respect.

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