Archive for MCMC convergence

Bayesian GANs [#2]

Posted in Books, pictures, R, Statistics with tags , , , , , , , , , , , , on June 27, 2018 by xi'an

As an illustration of the lack of convergence of the Gibbs sampler applied to the two “conditionals” defined in the Bayesian GANs paper discussed yesterday, I took the simplest possible example of a Normal mean generative model (one parameter) with a logistic discriminator (one parameter) and implemented the scheme (during an ISBA 2018 session). With flat priors on both parameters. And a Normal random walk as Metropolis-Hastings proposal. As expected, since there is no stationary distribution associated with the Markov chain, simulated chains do not exhibit a stationary pattern,

And they eventually reach an overflow error or a trapping state as the log-likelihood gets approximately to zero (red curve).

Too bad I missed the talk by Shakir Mohammed yesterday, being stuck on the Edinburgh by-pass at rush hour!, as I would have loved to hear his views about this rather essential issue…

convergences of MCMC and unbiasedness

Posted in pictures, Statistics, University life with tags , , , , , , , , , on January 16, 2018 by xi'an

During his talk on unbiased MCMC in Dauphine today, Pierre Jacob provided a nice illustration of the convergence modes of MCMC algorithms. With the stationary target achieved after 100 Metropolis iterations, while the mean of the target taking much more iterations to be approximated by the empirical average. Plus a nice connection between coupling time and convergence. Convergence to the target.During Pierre’s talk, some simple questions came to mind, from developing an “impatient user version”, as in perfect sampling, in order  to stop chains that run “forever”,  to optimising parallelisation in order to avoid problems of asynchronicity. While the complexity of coupling increases with dimension and the coupling probability goes down, the average coupling time varies but an unexpected figure is that the expected cost per iteration is of 2 simulations, irrespective of the chosen kernels. Pierre also made a connection with optimal transport coupling and stressed that the maximal coupling was for the proposal and not for the target.

ABC forecasts

Posted in Books, pictures, Statistics with tags , , , , , , , , on January 9, 2018 by xi'an

My friends and co-authors David Frazier, Gael Martin, Brendan McCabe, and Worapree Maneesoonthorn arXived a paper on ABC forecasting at the turn of the year. ABC prediction is a natural extension of ABC inference in that, provided the full conditional of a future observation given past data and parameters is available but the posterior is not, ABC simulations of the parameters induce an approximation of the predictive. The paper thus considers the impact of this extension on the precision of the predictions. And argues that it is possible that this approximation is preferable to running MCMC in some settings. A first interesting result is that using ABC and hence conditioning on an insufficient summary statistic has no asymptotic impact on the resulting prediction, provided Bayesian concentration of the corresponding posterior takes place as in our convergence paper under revision.

“…conditioning inference about θ on η(y) rather than y makes no difference to the probabilistic statements made about [future observations]”

The above result holds both in terms of convergence in total variation and for proper scoring rules. Even though there is always a loss in accuracy in using ABC. Now, one may think this is a direct consequence of our (and others) earlier convergence results, but numerical experiments on standard time series show the distinct feature that, while the [MCMC] posterior and ABC posterior distributions on the parameters clearly differ, the predictives are more or less identical! With a potential speed gain in using ABC, although comparing parallel ABC versus non-parallel MCMC is rather delicate. For instance, a preliminary parallel ABC could be run as a burnin’ step for parallel MCMC, since all chains would then be roughly in the stationary regime. Another interesting outcome of these experiments is a case when the summary statistics produces a non-consistent ABC posterior, but still leads to a very similar predictive, as shown on this graph.This unexpected accuracy in prediction may further be exploited in state space models, towards producing particle algorithms that are greatly accelerated. Of course, an easy objection to this acceleration is that the impact of the approximation is unknown and un-assessed. However, such an acceleration leaves room for multiple implementations, possibly with different sets of summaries, to check for consistency over replicates.

running ABC when the likelihood is available

Posted in Statistics with tags , , , , , on September 19, 2017 by xi'an

Today I refereed a paper where the authors used ABC to bypass convergence (and implementation) difficulties with their MCMC algorithm. And I am still pondering whether or not this strategy makes sense. If only because ABC needs to handle the same complexity and the same amount of parameters as an MCMC algorithm. While shooting “in the dark” by using the prior or a coarse substitute to the posterior. And I wonder at the relevance of simulating new data when the [true] likelihood value [at the observed data] can be computed. This would sound to me like the relevant and unique “statistics” worth considering…

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.

stability of noisy Metropolis-Hastings

Posted in Statistics with tags , , , , , , on September 28, 2016 by xi'an

noisymcmcFelipe Medina-Aguayo, Antony Lee and Gareth Roberts (all at Warwick University) have recently published—even though the paper was accepted a year ago—a paper in Statistics and Computing about a variant to the pseudo-marginal Metropolis-Hastings algorithm. The modification is to simulate an estimate of the likelihood or posterior at the current value of the Markov chain at every iteration, rather than reproducing the current estimate. The reason for this refreshment of the weight estimate is to prevent stickiness in the chain, when a random weight leads to a very high value of the posterior. Unfortunately, this change leads to a Markov chain with the wrong stationary distribution. When this stationary exists! The paper actually produces examples of transient noisy chains, even in simple cases such as a geometric target distribution. And even when taking the average of a large number of weights. But the paper also contains sufficient conditions, like negative weight moments or uniform ergodicity of the proposal, for the noisy chain to be geometrically ergodic. Even though the applicability of those conditions to complex targets is not always obvious.

MCMskv #5 [future with a view]

Posted in Kids, Mountains, R, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , on January 12, 2016 by xi'an

As I am flying back to Paris (with an afternoon committee meeting in München in-between), I am reminiscing on the superlative scientific quality of this MCMski meeting, on the novel directions in computational Bayesian statistics exhibited therein, and on the potential settings for the next meeting. If any.

First, as hopefully obvious from my previous entries, I found the scientific program very exciting, with almost uniformly terrific talks, and a coverage of the field of computational Bayesian statistics that is perfectly tuned to my own interest. In that sense, MCMski is my “top one” conference! Even without considering the idyllic location. While some of the talks were about papers I had already read (and commented here), others brought new vistas and ideas. If one theme is to emerge from this meeting it has to be the one of approximate and noisy algorithms, with a wide variety of solutions and approaches to overcome complexity issues. If anything, I wish the solutions would also incorporate the Boxian fact that the statistical models themselves are approximate. Overall, a fantastic program (says one member of the scientific committee).

Second, as with previous MCMski meetings, I again enjoyed the unique ambience of the meeting, which always feels more relaxed and friendly than other conferences of a similar size, maybe because of the après-ski atmosphere or of the special coziness provided by luxurious mountain hotels. This year hotel was particularly pleasant, with non-guests like myself able to partake of some of their facilities. A big thank you to Anto for arranging so meticulously all the details of such a large meeting!!! I am even more grateful when realising this is the third time Anto takes over the heavy load of organising MCMski. Grazie mille!

Since this is a [and even the!] BayesComp conference, the current section program chair and board must decide on the  structure and schedule of the next meeting. A few suggestions if I may: I would scrap entirely the name MCMski from the next conference as (a) it may sound like academic tourism for unaware bystanders (who only need to check the program of any of the MCMski conferences to stand reassured!) and (b) its topic go way beyond MCMC. Given the large attendance and equally large proportion of young researchers, I would also advise against hosting the conference in a ski resort for both cost and accessibility reasons [as we had already discussed after MCMskiv], in favour of a large enough town to offer a reasonable range of accommodations and of travel options. Like Chamonix, Innsbruck, Reykjavik, or any place with a major airport about one hour away… If nothing is available with skiing possibilities, so be it! While the outdoor inclinations of the early organisers induced us to pick locations where skiing over lunch break was a perk, any accessible location that allows for a concentration of researchers in a small area and for the ensuing day-long exchange is fine! Among the novelties in the program, the tutorials and the Breaking news! sessions were quite successful (says one member of the scientific committee). And should be continued in one format or another. Maybe a more programming thread could be added as well… And as we had mentioned earlier, to see a stronger involvement of the Young Bayesian section in the program would be great! (Even though the current meeting already had many young researcher  talks.)