Archive for Data augmentation

inefficiency of data augmentation for large samples

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , on May 31, 2016 by xi'an

On Monday, James Johndrow, Aaron Smith, Natesh Pillai, and David Dunson arXived a paper on the diminishing benefits of using data augmentation for large and highly imbalanced categorical data. They reconsider the data augmentation scheme of Tanner and Wong (1987), surprisingly not mentioned, used in the first occurrences of the Gibbs sampler like Albert and Chib’s (1993) or our mixture estimation paper with Jean Diebolt (1990). The central difficulty with data augmentation is that the distribution to be simulated operates on a space that is of order O(n), even when the original distribution covers a single parameter. As illustrated by the coalescent in population genetics (and the subsequent intrusion of the ABC methodology), there are well-known cases when the completion is near to impossible and clearly inefficient (as again illustrated by the failure of importance sampling strategies on the coalescent). The paper provides spectral gaps for the logistic and probit regression completions, which are of order a power of log(n) divided by √n, when all observations are equal to one. In a somewhat related paper with Jim Hobert and Vivek Roy, we studied the spectral gap for mixtures with a small number of observations: I wonder at the existence of a similar result in this setting, when all observations stem from one component of the mixture, when all observations are one. The result in this paper is theoretically appealing, the more because the posteriors associated with such models are highly regular and very close to Gaussian (and hence not that challenging as argued by Chopin and Ridgway). And because the data augmentation algorithm is uniformly ergodic in this setting (as we established with Jean Diebolt  and later explored with Richard Tweedie). As demonstrated in the  experiment produced in the paper, when comparing with HMC and Metropolis-Hastings (same computing times?), which produce much higher effective sample sizes.

corrected MCMC samplers for multivariate probit models

Posted in Books, pictures, R, Statistics, University life with tags , , , , , , , , on May 6, 2015 by xi'an

“Moreover, IvD point out an error in Nobile’s derivation which can alter its stationary distribution. Ironically, as we shall see, the algorithms of IvD also contain an error.”

 Xiyun Jiao and David A. van Dyk arXived a paper correcting an MCMC sampler and R package MNP for the multivariate probit model, proposed by Imai and van Dyk in 2005. [Hence the abbreviation IvD in the above quote.] Earlier versions of the Gibbs sampler for the multivariate probit model by Rob McCulloch and Peter Rossi in 1994, with a Metropolis update added by Agostino Nobile, and finally an improved version developed by Imai and van Dyk in 2005. As noted in the above quote, Jiao and van Dyk have discovered two mistakes in this latest version, jeopardizing the validity of the output.

IvDykThe multivariate probit model considered here is a multinomial model where the occurrence of the k-th category is represented as the k-th component of a (multivariate) normal (correlated) vector being the largest of all components. The latent normal model being non-identifiable since invariant by either translation or scale, identifying constraints are used in the literature. This means using a covariance matrix of the form Σ/trace(Σ), where Σ is an inverse Wishart random matrix. In their 2005 implementation, relying on marginal data augmentation—which essentially means simulating the non-identifiable part repeatedly at various steps of the data augmentation algorithm—, Imai and van Dyk missed a translation term and a constraint on the simulated matrices that lead to simulations outside the rightful support, as illustrated from the above graph [snapshot from the arXived paper].

IvDyk1Since the IvD method is used in many subsequent papers, it is quite important that these mistakes are signalled and corrected. [Another snapshot above shows how much both algorithm differ!] Without much thinking about this, I [thus idly] wonder why an identifying prior is not taking the place of a hard identifying constraint, as it should solve the issue more nicely. In that it would create less constraints and more entropy (!) in exploring the augmented space, while theoretically providing a convergent approximation of the identifiable parts. I may (must!) however miss an obvious constraint preventing this implementation.

recycling accept-reject rejections (#2)

Posted in R, Statistics, University life with tags , , , , , , on July 2, 2014 by xi'an

Following yesterday’s post on Rao’s, Liu’s, and Dunson’s paper on a new approach to intractable normalising constants, and taking advantage of being in Warwick, I tested the method on a toy model, namely the posterior associated with n Student’s t observations with unknown location parameter μ and a flat prior,

x_1,\ldots,x_n \sim p(x|\mu) \propto \left[ 1+(x-\mu)^2/\nu \right]^{-(\nu+1)/2}

which is “naturally” bounded by a Cauchy density with scale √ν. The constant M is then easily derived and running the new algorithm follows from a normal random walk proposal targeting the augmented likelihood (R code below).

dunsonAs shown by the above graph, the completion-by-rejection scheme produces a similar outcome (tomato) as the one based on the sole observations (steelblue). With a similar acceptance rate. However, the computing time is much much degraded:

> system.time(g8())
   user  system elapsed
 53.751   0.056  54.103
> system.time(g9())
   user  system elapsed
  1.156   0.000   1.161

when compared with the no-completion version. Here is the entire R code that produced both MCMC samples: Continue reading

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…

MCMC at ICMS (2)

Posted in Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on April 25, 2012 by xi'an

The second day of our workshop on computational statistics at the ICMS started with a terrific talk by Xiao-Li Meng. Although this talk related with his Inception talk in Paris last summer, and of the JCGS discussion paper, he brought new geometric aspects to the phenomenon (managing a zero correlation and hence i.i.d.-ness in the simulation of a Gaussian random effect posterior distribution). While I was reflecting about the difficulty to extend the perspective beyond normal models, he introduced a probit example where exact null correlation cannot be found but an adaptive scheme allows to explore the range of correlation coefficients. This made me somehow think of a possible version in this approach in a tempering perspective, where different data augmentation schemes would be merged into an “optimal” geometric mixture, rather than via interweaving.

As an aside, Xiao-Li mentioned the idea of Bayesian sufficiency and Bayesian ancilarity in the construction of his data augmentation schemes. He then concluded that sufficiency is identical in classical and Bayesian approaches, while ancilarity could be defined in several ways. I have already posted on that, but it seems to me that sufficiency is a weaker notion in the Bayesian perspective in the sense that all that matters is that the posterior is the same given the observation y and given the observed statistics, rather than uniformly over all possible values of the random variable Y as in the classical sense. As for ancilarity, it is also natural to consider that an ancillary statistics does not bring information on the parameter, i.e. that the prior and the posterior distributions are the same given the observed ancillary statistics. Going further to define ancilarity as posterior independence between “true” parameters and auxiliary variables, as Xiao-Li suggested, does not seem very sound as it leads to the paradoxes Basu liked so much!

Today, the overlap with the previous meetings in Bristol and in Banff was again limited: Arnaud Doucet rewrote his talk towards less technicity, which means I got the idea much more clearly than last week. The idea of having a sequence of pseudo-parameters with the same pseudo-prior seems to open a wide range of possible adaptive schemes. Faming Liang also gave a talk fairly similar to the one he presented in Banff. And David van Dyk as well, which led me to think anew about collapsed Gibbs samplers in connection with ABC and a project I just started here in Edinburgh.

Otherwise, the intense schedule of the day saw us through eleven talks. Daniele Impartato called for distributions (in the physics or Laurent Schwarz’ meaning of the term!) to decrease the variance of Monte Carlo estimations, an approach I hope to look further as Schwarz’ book is the first math book I ever bought!, an investment I tried to capitalize once in writing a paper mixing James-Stein estimation and distributions for generalised integration by part, paper that was repeatedly rejected until I gave up! Jim Griffin showed us improvements brought in the exploration of large number of potential covariates in linear and generalised linear models. Natesh Pillai tried to drag us through several of his papers on covariance matrix estimation, although I fear he lost me along the way! Let me perversely blame the schedule (rather than an early rise to run around Arthur’s Seat!) for falling asleep during Alex Beskos’ talk on Hamiltonian MCMC for diffusions, even though I was looking forward this talk. (Apologies to Alex!) Then Simon Byrne gave us a quick tour of differential geometry in connection with orthogonalization for Hamiltonian MCMC. Which brought me back very briefly to this early time I was still considering starting a PhD in differential geometry and then even more briefly played with the idea of mixing differential geometry and statistics à la Shun’ichi  Amari…. Ian Murray and  Simo Sarkka completed the day with a cartoonesque talk on latent Gaussians that connected well with Xiao-Li’s and a talk on Gaussian approximations to diffusions with unknown parameters, which kept within the main theme of the conference, namely inference on partly observed diffusions.

As written above, this was too intense a day, with hardly any free time to discuss about the talks or the ongoing projects, which makes me prefer the pace adopted in Bristol or in Banff. Having to meet a local student on leave from Dauphine for a year here did not help of course!)

Improving convergence properties of the Data Augmentation algorithm

Posted in Statistics, University life with tags , , , , on February 7, 2012 by xi'an

Our paper Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling, written with Jim Hobert and Vivek Roy during their latest visit to Paris a few years ago (!), has now appeared in arXiv, Statistical Science and project Euclid. I already mentioned in a previous post why this is an important paper for me. (There is nothing new  here, compared with this earlier post, except that the paper was re-posted on arXiv!)

Improving convergence of Data Augmentation [published]

Posted in Statistics with tags , , , , on November 4, 2011 by xi'an

Our paper with Jim Hobert and Vivek Roy, Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling, has now appeared in Statistical Science and is available on Project Euclid. (For IMS members, at least.) Personally, this is an important paper, not only for providing an exact convergence evaluation for mixtures,  not only for sharing exciting research days with my friends Jim and Vivek, but also for finalising a line of research somehow started in 1993 when Richard Tweedie visited me in Paris and when I visited him in Fort Collins… Coincidentally, my discussion of Don Fraser’s provocative Is Bayes Posterior just Quick and Dirty Confidence? also appeared in this issue of Statistical Science.