one bridge further

Jackie Wong, Jon Forster (Warwick) and Peter Smith have just published a paper in Statistics & Computing on bridge sampling bias and improvement by splitting.

“… known to be asymptotically unbiased, bridge sampling technique produces biased estimates in practical usage for small to moderate sample sizes (…) the estimator yields positive bias that worsens with increasing distance between the two distributions. The second type of bias arises when the approximation density is determined from the posterior samples using the method of moments, resulting in a systematic underestimation of the normalizing constant.”

Recall that bridge sampling is based on a double trick with two samples x and y from two (unnormalised) densities f and g that are interverted in a ratio

of unbiased estimators of the inverse normalising constants. Hence biased. The more the less similar these two densities are. Special cases for ω include importance sampling [unbiased] and reciprocal importance sampling. Since the optimal version of the bridge weight ω is the inverse of the mixture of f and g, it makes me wonder at the performance of using both samples top and bottom, since as an aggregated sample, they also come from the mixture, as in Owen & Zhou (2000) multiple importance sampler. However, a quick try with a positive Normal versus an Exponential with rate 2 does not show an improvement in using both samples top and bottom (even when using the perfectly normalised versions)

morc=(sum(f(y)/(nx*dnorm(y)+ny*dexp(y,2)))+
            sum(f(x)/(nx*dnorm(x)+ny*dexp(x,2))))/(
  sum(g(x)/(nx*dnorm(x)+ny*dexp(x,2)))+
         sum(g(y)/(nx*dnorm(y)+ny*dexp(y,2))))

at least in terms of bias… Surprisingly (!) the bias almost vanishes for very different samples sizes either in favour of f or in favour of g. This may be a form of genuine defensive sampling, who knows?! At the very least, this ensures a finite variance for all weights. (The splitting approach introduced in the paper is a natural solution to create independence between the first sample and the second density. This reminded me of our two parallel chains in AMIS.)

EntropyMCMC [R package]

My colleague from the Université d’Orléans, Didier Chauveau, has just published on CRAN a new R package called EntropyMCMC, which contains convergence assessment tools for MCMC algorithms, based on non-parametric estimates of the Kullback-Leibler divergence between current distribution and target. (A while ago, quite a while ago!, we actually collaborated with a few others on the Springer-Verlag Lecture Note #135 Discretization and MCMC convergence assessments.) This follows from a series of papers by Didier Chauveau and Pierre Vandekerkhove that started with a nearest neighbour entropy estimate. The evaluation of this entropy is based on N iid (parallel) chains, which involves a parallel implementation. While the missing normalising constant is overwhelmingly unknown, the authors this is not a major issue “since we are mostly interested in the stabilization” of the entropy distance. Or in the comparison of two MCMC algorithms. [Disclaimer: I have not experimented with the package so far, hence cannot vouch for its performances over large dimensions or problematic targets, but would as usual welcome comments and feedback on readers’ experiences.]

call for sessions and labs at Bay2sC0mp²⁰

A call to all potential participants to the incoming BayesComp 2020 conference at the University of Florida in Gainesville, Florida, 7-10 January 2020, to submit proposals [to me] for contributed sessions on everything computational or training labs [to David Rossell] on a specific language or software. The deadline is April 1 and the sessions will be selected by the scientific committee, other proposals being offered the possibility to present the associated research during a poster session [which always is a lively component of the conference]. (Conversely, we reserve the possibility of a “last call” session made from particularly exciting posters on new topics.) Plenary speakers for this conference are

• David Blei (Columbia University)
• Paul Fearnhead (University of Lancaster)
• Emily Fox (University of Washington)
• Max Welling (University of Amsterdam)

and the first invited sessions are already posted on the webpage of the conference. We dearly hope to attract a wide area of research interests into a as diverse as possible program, so please accept this invitation!!!

Implicit maximum likelihood estimates

An ‘Og’s reader pointed me to this paper by Li and Malik, which made it to arXiv after not making it to NIPS. While the NIPS reviews were not particularly informative and strongly discordant, the authors point out in the comments that they are available for the sake of promoting discussion. (As made clear in earlier posts, I am quite supportive of this attitude! Disclaimer: I was not involved in an evaluation of this paper, neither for NIPS nor for another conference or journal!!) Although the paper does not seem to mention ABC in the setting of implicit likelihoods and generative models, there is a reference to the early (1984) paper by Peter Diggle and Richard Gratton that is often seen as the ancestor of ABC methods. The authors point out numerous issues with solutions proposed for parameter estimation in such implicit models. For instance, for GANs, they signal that “minimizing the Jensen-Shannon divergence or the Wasserstein distance between the empirical data distribution and the model distribution does not necessarily minimize the same between the true data distribution and the model distribution.” (Not mentioning the particular difficulty with Bayesian GANs.) Their own solution is the implicit maximum likelihood estimator, which picks the value of the parameter θ bringing a simulated sample the closest to the observed sample. Closest in the sense of the Euclidean distance between both samples. Or between the minimum of several simulated samples and the observed sample. (The modelling seems to imply the availability of n>1 observed samples.) They advocate using a stochastic gradient descent approach for finding the optimal parameter θ which presupposes that the dependence between θ and the simulated samples is somewhat differentiable. (And this does not account for using a min, which would make differentiation close to impossible.) The paper then meanders in a lengthy discussion as to whether maximising the likelihood makes sense, with a rather naïve view on why using the empirical distribution in a Kullback-Leibler divergence does not make sense! What does not make sense is considering the finite sample approximation to the Kullback-Leibler divergence with the true distribution in my opinion.

approximate likelihood

Today, I read a newly arXived paper by Stephen Gratton on a method called GLASS for General Likelihood Approximate Solution Scheme… The starting point is the same as with ABC or synthetic likelihood, namely a collection of summary statistics and an intractable likelihood. The author proposes to use as a substitute a maximum entropy solution based on these summary statistics and their assumed moments under the theoretical model. What is quite unclear in the paper is whether or not these assumed moments are available in closed form or not. Otherwise, it would appear as a variant to the synthetic likelihood [aka simulated moments] approach, meaning that the expectations of the summary statistics under the theoretical model and for a given value of the parameter are obtained through Monte Carlo approximations. (All the examples therein allow for closed form expressions.)