## approximation of Bayes Factors via mixing

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on December 21, 2020 by xi'an

A [new version of a] paper by Chenguang Dai and Jun S. Liu got my attention when it appeared on arXiv yesterday. Due to its title which reminded me of a solution to the normalising constant approximation that we proposed in the 2010 nested sampling evaluation paper we wrote with Nicolas. Recovering bridge sampling—mentioned by Dai and Liu as an alternative to their approach rather than an early version—by a type of Charlie Geyer (1990-1994) trick. (The attached slides are taken from my MCMC graduate course, with a section on the approximation of Bayesian normalising constants I first wrote for a short course at Jim Berger’s 70th anniversary conference, in San Antonio.)

A difference with the current paper is that the authors “form a mixture distribution with an adjustable mixing parameter tuned through the Wang-Landau algorithm.” While we chose it by hand to achieve sampling from both components. The weight is updated by a simple (binary) Wang-Landau version, where the partition is determined by which component is simulated, ie by the mixture indicator auxiliary variable. Towards using both components on an even basis (à la Wang-Landau) and stabilising the resulting evaluation of the normalising constant. More generally, the strategy applies to a sequence of surrogate densities, which are chosen by variational approximations in the paper.

## marginal likelihood with large amounts of missing data

Posted in Books, pictures, Statistics with tags , , , , , , , , on October 20, 2020 by xi'an

In 2018, Panayiota Touloupou, research fellow at Warwick, and her co-authors published a paper in Bayesian analysis that somehow escaped my radar, despite standing in my first circle of topics of interest! They construct an importance sampling approach to the approximation of the marginal likelihood, the importance function being approximated from a preliminary MCMC run, and consider the special case when the sampling density (i.e., the likelihood) can be represented as the marginal of a joint density. While this demarginalisation perspective is rather usual, the central point they make is that it is more efficient to estimate the sampling density based on the auxiliary or latent variables than to consider the joint posterior distribution of parameter and latent in the importance sampler. This induces a considerable reduction in dimension and hence explains (in part) why the approach should prove more efficient. Even though the approximation itself is costly, at about 5 seconds per marginal likelihood. But a nice feature of the paper is to include the above graph that includes both computing time and variability for different methods (the blue range corresponding to the marginal importance solution, the red range to RJMCMC and the green range to Chib’s estimate). Note that bridge sampling does not appear on the picture but returns a variability that is similar to the proposed methodology.

## one bridge further

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , on June 30, 2020 by xi'an

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

$m \sum_{i=1}^n g(x_i)\omega(x_i) \Big/ n \sum_{i=1}^m f(y_i)\omega(y_i)$

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.)

## Mallows model with intractable constant

Posted in Books, pictures, Statistics with tags , , , , , , , , on November 21, 2019 by xi'an

The paper Probabilistic Preference Learning with the Mallows Rank Model by Vitelli et al. was published last year in JMLR which may be why I missed it. It brings yet another approach to the perpetual issue of intractable  normalising constants. Here, the data is made of rankings of n objects by N experts, with an assumption of a latent ordering ρ acting as “mean” in the Mallows model. Along with a scale α, both to be estimated, and indeed involving an intractable normalising constant in the likelihood that only depends on the scale α because the distance is right-invariant. For instance the Hamming distance used in coding. There exists a simplification of the expression of the normalising constant due to the distance only taking a finite number of values, multiplied by the number of cases achieving a given value. Still this remains a formidable combinatoric problem. Running a Gibbs sampler is not an issue for the parameter ρ as the resulting Metropolis-Hastings-within-Gibbs step does not involve the missing constant. But it poses a challenge for the scale α, because the Mallows model cannot be exactly simulated for most distances. Making the use of pseudo-marginal and exchange algorithms presumably impossible. The authors use instead an importance sampling approximation to the normalising constant relying on a pseudo-likelihood version of Mallows model and a massive number (10⁶ to 10⁸) of simulations (in the humongous set of N-sampled permutations of 1,…,n). The interesting point in using this approximation is that the convergence result associated with pseudo-marginals no long applies and that the resulting MCMC algorithm converges to another limiting distribution. With the drawback that this limiting distribution is conditional to the importance sample. Various extensions are found in the paper, including a mixture of Mallows models. And an round of applications, including one on sushi preferences across Japan (fatty tuna coming almost always on top!). As the authors note, a very large number of items like n>10⁴ remains a challenge (or requires an alternative model).

## revisiting the balance heuristic

Posted in Statistics with tags , , , , , , , on October 24, 2019 by xi'an

Last August, Felipe Medina-Aguayo (a former student at Warwick) and Richard Everitt (who has now joined Warwick) arXived a paper on multiple importance sampling (for normalising constants) that goes “exploring some improvements and variations of the balance heuristic via a novel extended-space representation of the estimator, leading to straightforward annealing schemes for variance reduction purposes”, with the interesting side remark that Rao-Blackwellisation may prove sub-optimal when there are many terms in the proposal family, in the sense that not every term in the mixture gets sampled. As already noticed by Victor Elvira and co-authors, getting rid of the components that are not used being an improvement without inducing a bias. The paper also notices that the loss due to using sample sizes rather than expected sample sizes is of second order, compared with the variance of the compared estimators. It further relates to a completion or auxiliary perspective that reminds me of the approaches we adopted in the population Monte Carlo papers and in the vanilla Rao-Blackwellisation paper. But it somewhat diverges from this literature when entering a simulated annealing perspective, in that the importance distributions it considers are freely chosen as powers of a generic target. It is quite surprising that, despite the normalising weights being unknown, a simulated annealing approach produces an unbiased estimator of the initial normalising constant. While another surprise therein is that the extended target associated to their balance heuristic does not admit the right density as marginal but preserves the same normalising constant… (This paper will be presented at BayesComp 2020.)