Archive for MCMC

next BayesComp conference planned for Jan 2018, any volunteer?

Posted in Kids, Mountains, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , on February 25, 2016 by xi'an

MCMSki III poster, 2010 (C.) IMS[A call from the BayesComp section of ISBA for the next Bayesian computation meeting! As suggested in an earlier post, the label MCMski is discontinued to allow for any location amenable to organise a 200 plus meeting in good and hopefully reasonably priced conditions.]

The Bayesian Computation Section of ISBA is soliciting proposals to host its flagship meeting: BayesComp 2018

The expectation is that the meeting will be held in January 2018, but the committee will consider proposals for other times through January 2019. This meeting is a continuation of the popular MCMSki on recent advances in the theory and application of Bayesian computational methods such as MCMC. The tradition was to hold MCMski meetings in ski resorts, but, as the name change suggests, we encourage applications from any venue that could support BC2018.

A three-day meeting is planned, perhaps with an additional day or two of satellite meetings and/or short courses. One page proposals should address feasibility of hosting the meeting including

1. Proposed dates.
2. Transportation for international participants (both the proximity of international airports and transportation to/from the venue).
3. The conference facilities.
4. The availability and cost of hotels, including low cost options.
5. The proposed local organizing committee and their collective experience organizing international meetings.
6. Expected or promised contributions from the host organization, host country, or industrial partners towards the cost of running the meetings.

Proposals should be submitted to Nicolas Chopin (Program Chair) no later than May 31, 2016. The Board of Bayesian Computing Section will evaluate the proposals, choose a venue, and appoint the Program Committee for BayesComp 2018.

patterns of scalable Bayesian inference

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on February 24, 2016 by xi'an

Elaine Angelino, Matthew Johnson and Ryan Adams just arXived a massive survey of 118 pages on scalable Bayesian inference, which could have been entitled Bayes for Big Data, as this monograph covers state-of-the-art computational approaches to large and complex data structures. I did not read each and every line of it, but I have already recommended it to my PhD students. Some of its material unsurprisingly draws from the recent survey by Rémi Bardenet et al. (2015) I discussed a while ago. It also relates rather frequently to the somewhat parallel ICML paper of Korattikara et al. (2014). And to the firefly Monte Carlo procedure also discussed previously here.

Chapter 2 provides some standard background on computational techniques, Chapter 3 covers MCMC with data subsets, Chapter 4 gives some entries on MCMC with parallel and distributed architectures, Chapter 5 focus on variational solutions, and Chapter 6 is about open questions and challenges.

“Insisting on zero asymptotic bias from Monte Carlo estimates of expectations may leave us swamped in errors from high variance or transient bias.”

One central theme of the paper is the need for approximate solutions, MCMC being perceived as the exact solution. (Somewhat wrongly in the sense that the product of an MCMC is at best an empirical version of the true posterior, hence endowed with a residual and incompressible variation for a given computing budget.) While Chapter 3 stresses the issue of assessing the distance to the true posterior, it does not dwell at all on computing times and budget, which is arguably a much harder problem. Chapter 4 seems to be more aware of this issue since arguing that “a way to use parallel computing resources is to run multiple sequential MCMC algorithms at once [but that this] does not reduce the transient bias in MCMC estimates of posterior expectations” (p.54). The alternatives are to use either prefetching (which was the central theme of Elaine Angelino’s thesis), asynchronous Gibbs with the new to me (?) Hogwild Gibbs algorithms (connected in Terenin et al.’s recent paper, not quoted in the paper), some versions of consensus Monte Carlo covered in earlier posts, the missing links being in my humble opinion an assessment of the worth of those solutions (in the spirit of “here’s the solution, what was the problem again?”) and once again the computing time issue. Chapter 5 briefly discusses some recent developments in variational mean field approximations, which is farther from my interests and (limited) competence, but which appears as a particular class of approximate models and thus could (and should?) relate to likelihood-free methods. Chapter 6 about the current challenges of the field is presumably the most interesting in this monograph in that it produces open questions and suggests directions for future research. For instance, opposing the long term MCMC error with the short term transient part. Or the issue of comparing different implementations in a practical and timely perspective.

high dimension Metropolis-Hastings algorithms

Posted in Books, Kids, Mountains, pictures, R, Statistics with tags , , , , , , on January 26, 2016 by xi'an

When discussing high dimension models with Ingmar Schüster Schuster [blame my fascination for accented characters!] the other day, we came across the following paradox with Metropolis-Hastings algorithms. If attempting to simulate from a multivariate standard normal distribution in a large dimension, when starting from the mode of the target, i.e., its mean γ, leaving the mode γis extremely unlikely, given the huge drop between the value of the density at the mode γ and at likely realisations (corresponding to the blue sequence). Even when relying on the very scale that makes the proposal identical to the target! Resorting to a tiny scale like Σ/p manages to escape the unhealthy neighbourhood of the highly unlikely mode (as shown with the brown sequence).

Here is the corresponding R code:

p=100
T=1e3
mh=mu #mode as starting value
vale=rep(0,T)
for (t in 1:T){
prop=mvrnorm(1,mh,sigma/p)
if (log(runif(1))<logdmvnorm(prop,mu,sigma)-
   logdmvnorm(mh,mu,sigma)) mh=prop
vale[t]=logdmvnorm(mh,mu,sigma)}

Je reviendrai à Montréal [D-2]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on December 9, 2015 by xi'an

I have spent the day and more completing and compiling slides for my contrapuntal perspective on probabilistic numerics, back in Montréal, for the NIPS 2015 workshop of December 11 on this theme. As I presume the kind  invitation by the organisers was connected with my somewhat critical posts on the topic, I mostly  The day after, while I am flying back to London for the CFE (Computational and Financial Econometrics) workshop, somewhat reluctantly as there will be another NIPS workshop that day on scalable Monte Carlo.

Je veux revoir le long désert
Des rues qui n’en finissent pas
Qui vont jusqu’au bout de l’hiver
Sans qu’il y ait trace de pas

multiple importance sampling

Posted in Books, Statistics, University life with tags , , , , , , , , on November 20, 2015 by xi'an

“Within this unified context, it is possible to interpret that all the MIS algorithms draw samples from a equal-weighted mixture distribution obtained from the set of available proposal pdfs.”

In a very special (important?!) week for importance sampling!, Elvira et al. arXived a paper about generalized multiple importance sampling. The setting is the same as in earlier papers by Veach and Gibas (1995) or Owen and Zhou (2000) [and in our AMIS paper], namely a collection of importance functions and of simulations from those functions. However, there is no adaptivity for the construction of the importance functions and no Markov (MCMC) dependence on the generation of the simulations.

multipl
“One of the goals of this paper is to provide the practitioner with solid theoretical results about the superiority of some specific MIS schemes.”

One first part deals with the fact that a random point taken from the conjunction of those samples is distributed from the equiweighted mixture. Which was a fact I had much appreciated when reading Owen and Zhou (2000). From there, the authors discuss the various choices of importance weighting. Meaning the different degrees of Rao-Blackwellisation that can be applied to the sample. As we discovered in our population Monte Carlo research [which is well-referred within this paper], conditioning too much leads to useless adaptivity. Again a sort of epiphany for me, in that a whole family of importance functions could be used for the same target expectation and the very same simulated value: it all depends on the degree of conditioning employed for the construction of the importance function. To get around the annoying fact that self-normalised estimators are never unbiased, the authors borrow Liu’s (2000) notion of proper importance sampling estimators, where the ratio of the expectations is returning the right quantity. (Which amounts to recover the correct normalising constant(s), I believe.) They then introduce five (5!) different possible importance weights that all produce proper estimators. However, those weights correspond to different sampling schemes, so do not apply to the same sample. In other words, they are not recycling weights as in AMIS. And do not cover the adaptive cases where the weights and parameters of the different proposals change along iterations. Unsurprisingly, the smallest variance estimator is the one based on sampling without replacement and an importance weight made of the entire mixture. But this result does not apply for the self-normalised version, whose variance remains intractable.

I find this survey of existing and non-existing multiple importance methods quite relevant and a must-read for my students (and beyond!). My reservations (for reservations there must be!) are that the study stops short of pushing further the optimisation. Indeed, the available importance functions are not equivalent in terms of the target and hence weighting them equally is sub-efficient. The adaptive part of the paper broaches upon this issue but does not conclude.

data augmentation with divergence

Posted in Books, Kids, Statistics, University life with tags , , , , , on November 18, 2015 by xi'an

Another (!) Cross Validated question that shed some light on the difficulties of explaining the convergence of MCMC algorithms. Or in understanding conditioning and hierarchical models. The author wanted to know why a data augmentation of his did not converge: In a simplified setting, given an observation y that he wrote as y=h(x,θ), he had built a Gibbs sampler by reconstructing x=g(y,θ) and simulating θ given x: at each iteration t,

  1. compute xt=g(y,θt-1)
  2. simulate θt~π(θ|xt)

and he attributed the lack of convergence to a possible difficulty with the Jacobian. My own interpretation of the issue was rather that condition on the unobserved x was not the same as conditioning on the observed y and hence that y was missing from step 2. And that the simulation of x is useless. Unless one uses it in an augmented scheme à la Xiao-Li… Nonetheless, I like the problem, if only because my very first reaction was to draw a hierarchical dependence graph and to conclude this should be correct, before checking on a toy example that it was not!

adaptive and delayed MCMC for expensive likelihoods [reply from the authors]

Posted in Books, Statistics with tags , , , , , , , , on October 29, 2015 by xi'an

[Chris Sherlock, Andrew Golightly and Daniel Henderson have written a reply about my earlier comments on their arXived paper which works better as a post than as a comment:]

Thank you for the constructive criticism of our paper. Our approach uses a simple weighted average of nearest neighbours and we agree that GPs offer a useful alternative. Both methods have pros and cons, however we first note a similarity: Kriging using a GP also leads to a weighted average of values.

The two most useful pros of the GP are that, (i) by estimating the parameters of the GP one may represent the scales of variability more accurately than a simple nearest neighbour approach with weighting according to Euclidean distance, and (ii) one obtains a distribution for the uncertainty in the Kriging estimate of the log-likelihood.

Both the papers in the blog entry (as well as other recent papers which use GPs), in one way or another take advantage of the second point. However, as acknowledged in Richard Wilkinson’s paper, estimating the parameters of a GP is computationally very costly, and this estimation must be repeated as the training data set grows. Probably for this reason and because of the difficulty in identifying p(p+1)/2 kernel range parameters, Wilkinson’s paper uses a diagonal covariance structure for the kernel. We can find no description of the structure of the covariance function that is used for each statistic in the Meeds & Welling paper but this issue is difficult to avoid.

Our initial training run is used to transform the parameters so that they are approximately orthogonal with unit variance and Euclidean distance is a sensible metric. This has two consequences: (i) the KD-tree is easier to set up and use, and (ii) the nearest neighbours in a KD-tree that is approximately balanced can be found in O(log N) operations, where N is the number of training points. Both (i) and (ii) only require Euclidean distance to be a reasonable measure, not perfect, so there is no need for the training run to have “properly converged”, just for it to represent the gross relationships in the posterior and for the transformation to be 1-1. We note a parallel between our approximate standardisation using training data, and the need to estimate a symmetric matrix of distance parameters from training data to obtain a fully representative GP kernel.

The GP approach might lead to a more accurate estimate of the posterior than a nearest neighbour approach (for a fixed number of training points), but this is necessary for the algorithms in the papers mentioned above since they sample from an approximation to the posterior. As noted in the blog post the delayed-acceptance step (which also could be added to GP-based algorithms) ensures that our algorithm samples from the true posterior so accuracy is helpful for efficiency rather than essential for validity.

We have made the kd-tree C code available and put some effort into making the interface straightforward to use. Our starting point is an existing simple MCMC algorithm; as it is already evaluating the posterior (or an unbiased approximation) then why not store this and take advantage of it within the existing algorithm? We feel that our proposal offers a relatively cheap and straightforward route for this.

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