Archive for convergence of Gibbs samplers

Gibbs sampling with incompatible conditionals

Posted in Books, Kids, R, Statistics with tags , , , , , , on July 23, 2019 by xi'an

An interesting question (with no clear motivation) on X validated wondering why a Gibbs sampler produces NAs… Interesting because multi-layered:

  1. The attached R code indeed produces NAs because it calls the Negative Binomial Neg(x¹,p) random generator with a zero success parameter, x¹=0, which automatically returns NAs. This can be escaped by returning a one (1) instead.
  2. The Gibbs sampler is based on a Bin(x²,p) conditional for X¹ and a Neg(x¹,p) conditional for X². When using the most standard version of the Negative Binomial random variate as the number of failures, hence supported on 0,1,2…. these two conditionals are incompatible, i.e., there cannot be a joint distribution behind that returns these as conditionals, which makes the limiting behaviour of the Markov chain harder to study. It however seems to converge to a distribution close to zero, which is not contradictory with the incompatibility property: the stationary joint distribution simply does not enjoy the conditionals used by the Gibbs sampler as its conditionals.
  3. When using the less standard version of the Negative Binomial random variate understood as a number of attempts for the conditional on X², the two conditionals are compatible and correspond to a joint measure proportional to x_1^{-1} {x_1 \choose x_2} p^{x_2} (1-p)^{x_1-x_2}, however this pmf does not sum up to a finite quantity (as in the original Gibbs for Kids example!), hence the resulting Markov chain is at best null recurrent, which seems to be the case for p different from ½. This is unclear to me for p=½.

automatic adaptation of MCMC algorithms

Posted in pictures, Statistics with tags , , , , , , , on March 4, 2019 by xi'an

“A typical adaptive MCMC sampler will approximately optimize performance given the kind of sampler chosen in the first place, but it will not optimize among the variety of samplers that could have been chosen.”

Last February (2018), Dao Nguyen and five co-authors arXived a paper that I missed. On a new version of adaptive MCMC that aims at selecting a wider range of proposal kernels. Still requiring a by-hand selection of this collection of kernels… Among the points addressed, beyond the theoretical guarantees that the adaptive scheme does not jeopardize convergence to the proper target, are a meta-exploration of the set of combinations of samplers and integration of the computational speed in the assessment of each sampler. Including the very difficulty of assessing mixing. One could deem the index of the proposal as an extra (cyber-)parameter to its generic parameter (like the scale in the random walk), but the discreteness of this index makes the extension more delicate than expected. And justifies the distinction between internal and external parameters. The notion of a worst-mixing dimension is quite appealing and connects with the long-term hope that one could spend the maximum fraction of the sampler runtime over the directions that are poorly mixing, while still keeping the target as should be. The adaptive scheme is illustrated on several realistic models with rather convincing gains in efficiency and time.

The convergence tools are inspired from Roberts and Rosenthal (2007), with an assumption of uniform ergodicity over all kernels considered therein which is both strong and delicate to assess in practical settings. Efficiency is rather unfortunately defined in terms of effective sample size, which is a measure of correlation or lack thereof, but which does not relate to the speed of escape from the basin of attraction of the starting point. I also wonder at the pertinence of the estimation of the effective sample size when the chain is based on different successive kernels, since the lack of correlation may be due to another kernel. Another calibration issue is the internal clock that relates to the average number of iterations required to tune properly a specific kernel, which again may be difficult to assess in a realistic situation. A last query is whether or not this scheme could be compared with an asynchronous (and valid) MCMC approach that exploits parallel capacities of the computer.

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!

Bayesian gan [gan style]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , on June 26, 2018 by xi'an

In their paper Bayesian GANS, arXived a year ago, Saatchi and Wilson consider a Bayesian version of generative adversarial networks, putting priors on both the model and the discriminator parameters. While the prospect seems somewhat remote from genuine statistical inference, if the following statement is representative

“GANs transform white noise through a deep neural network to generate candidate samples from a data distribution. A discriminator learns, in a supervised manner, how to tune its parameters so as to correctly classify whether a given sample has come from the generator or the true data distribution. Meanwhile, the generator updates its parameters so as to fool the discriminator. As long as the generator has sufficient capacity, it can approximate the cdf inverse-cdf composition required to sample from a data distribution of interest.”

I figure the concept can also apply to a standard statistical model, where x=G(z,θ) rephrases the distributional assumption x~F(x;θ) via a white noise z. This makes resorting to a prior distribution on θ more relevant in the sense of using potential prior information on θ (although the successes of probabilistic numerics show formal priors can be used on purely numerical ground).

The “posterior distribution” that is central to the notion of Bayesian GANs is however unorthodox in that the distribution is associated with the following conditional posteriors

where D(x,θ) is the “discriminator”, that is, in GAN lingo, the probability to be allocated to the “true” data generating mechanism rather than to the one associated with G(·,θ). The generative conditional posterior (1) then aims at fooling the discriminator, i.e. favours generative parameter values that raise the probability of wrong allocation of the pseudo-data. The discriminative conditional posterior (2) is a standard Bayesian posterior based on the original sample and the generated sample. The authors then iteratively sample from these posteriors, effectively implementing a two-stage Gibbs sampler.

“By iteratively sampling from (1) and (2) at every step of an epoch one can, in the limit, obtain samples from the approximate posteriors over [both sets of parameters].”

What worries me about this approach is that  just cannot work, in the sense that (1) and (2) cannot be compatible conditional (posterior) distributions. There is no joint distribution for which (1) and (2) would be the conditionals, since the pseudo-data appears in D for (1) and (1-D) in (2). This means that the convergence of a Gibbs sampler is at best to a stationary σ-finite measure. And hence that the meaning of the chain is delicate to ascertain… Am I missing any fundamental point?! [I checked the reviews on NIPS webpage and could not spot this issue being raised.]

Gibbs for kidds

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , on February 12, 2018 by xi'an

 

A chance (?) question on X validated brought me to re-read Gibbs for Kids, 25 years after it was written (by my close friends George and Ed). The originator of the question had difficulties with the implementation, apparently missing the cyclic pattern of the sampler, as in equations (2.3) and (2.4), and with the convergence, which is only processed for a finite support in the American Statistician paper. The paper [which did not appear in American Statistician under this title!, but inspired an animal bredeer, Dan Gianola, to write a “Gibbs for pigs” presentation in 1993 at the 44th Annual Meeting of the European Association for Animal Production, Aarhus, Denmark!!!] most appropriately only contains toy examples since those can be processed and compared to know stationary measures. This is for instance the case for the auto-exponential model

f(x,y) \propto exp(-xy)

which is only defined as a probability density for a compact support. (The paper does not identify the model as a special case of auto-exponential model, which apparently made the originator of the model, Julian Besag in 1974, unhappy, as George and I found out when visiting Bath, where Julian was spending the final year of his life, many years later.) I use the limiting case all the time in class to point out that a Gibbs sampler can be devised and operate without a stationary probability distribution. However, being picky!, I would like to point out that, contrary, to a comment made in the paper, the Gibbs sampler does not “fail” but on the contrary still “converges” in this case, in the sense that a conditional ergodic theorem applies, i.e., the ratio of the frequencies of visits to two sets A and B with finite measure do converge to the ratio of these measures. For instance, running the Gibbs sampler 10⁶ steps and ckecking for the relative frequencies of x’s in (1,2) and (1,3) gives 0.685, versus log(2)/log(3)=0.63, since 1/x is the stationary measure. One important and influential feature of the paper is to stress that proper conditionals do not imply proper joints. George would work much further on that topic, in particular with his PhD student at the time, my friend Jim Hobert.

With regard to the convergence issue, Gibbs for Kids points out to Schervish and Carlin (1990), which came quite early when considering Gelfand and Smith published their initial paper the very same year, but which also adopts a functional approach to convergence, along the paper’s fixed point perspective, somehow complicating the matter. Later papers by Tierney (1994), Besag (1995), and Mengersen and Tweedie (1996) considerably simplified the answer, which is that irreducibility is a necessary and sufficient condition for convergence. (Incidentally, the reference list includes a technical report of mine’s on latent variable model MCMC implementation that never got published.)

a new paradigm for improper priors

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , on November 6, 2017 by xi'an

Gunnar Taraldsen and co-authors have arXived a short note on using improper priors from a new perspective. Generalising an earlier 2016 paper in JSPI on the same topic. Which both relate to a concept introduced by Rényi (who himself attributes the idea to Kolmogorov). Namely that random variables measures are to be associated with arbitrary measures [not necessarily σ-finite measures, the later defining σ-finite random variables], rather than those with total mass one. Which allows for an alternate notion of conditional probability in the case of σ-finite random variables, with the perk that this conditional probability distribution is itself of mass 1 (a.e.).  Which we know happens when moving from prior to proper posterior.

I remain puzzled by the 2016 paper though as I do not follow the meaning of a random variable associated with an infinite mass probability measure. If the point is limited to construct posterior probability distributions associated with improper priors, there is little value in doing so. The argument in the 2016 paper is however that one can then define a conditional distribution in marginalisation paradoxes à la Stone, Dawid and Zidek (1973) where the marginal does not exist. Solving with this formalism the said marginalisation paradoxes as conditional distributions are only defined for σ-finite random variables. Which gives a fairly different conclusion from either Stone, Dawid and Zidek (1973) [with whom I agree, namely that there is no paradox because there is no “joint” distribution] or Jaynes (1973) [with whom I less agree!, in that the use of an invariant measure to make the discrepancy go away is not a particularly strong argument in favour of this measure]. The 2016 paper also draws an interesting connection with the study by Jim Hobert and George Casella (in Jim’s thesis) of [null recurrent or transient] Gibbs samplers with no joint [proper] distribution. Which in some situations can produce proper subchains, a phenomenon later exhibited by Alan Gelfand and Sujit Sahu (and Xiao-Li Meng as well if I correctly remember!). But I see no advantage in following this formalism, as it does not impact whether the chain is transient or null recurrent, or anything connected with its implementation. Plus a link to the approximation of improper priors by sequences of proper ones by Bioche and Druihlet I discussed a while ago.

pseudo-marginal MCMC with minimal replicas

Posted in Books, Statistics, University life with tags , , , , , , on November 16, 2016 by xi'an

A week ago, Chris Sherlock, Alexandre Thiery and Anthony Lee (Warwick) arXived a short paper where they show that it is most efficient to use only one random substitute to the likelihood when considering a pseudo-marginal algorithm. Thus generalising an earlier paper of Luke Bornn and co-authors I commented earlier. A neat side result in the paper is that the acceptance probability for m replicas [in the pseudo-marginal approximation] is at most m/s the acceptance probability for s replicas when s<m. The main result is as follows:

screenshot_20161114_140345There is a (superficial?) connection with the fact that when running Metropolis-within-Gibbs there is no advantage in doing more than one single Metropolis iteration, as the goal is to converge to the joint posterior, rather than approximating better the full conditional…