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.)

nested sampling via SMC

“We show that by implementing a special type of [sequential Monte Carlo] sampler that takes two im-portance sampling paths at each iteration, one obtains an analogous SMC method to [nested sampling] that resolves its main theoretical and practical issues.”

A paper by Queenslander Robert Salomone, Leah South, Chris Drovandi and Dirk Kroese that I had missed (and recovered by Grégoire after we discussed this possibility with our Master students). On using SMC in nested sampling. What are the difficulties mentioned in the above quote?

1. Dependence between the simulated samples, since only the offending particle is moved by one or several MCMC steps. (And MultiNest is not a foolproof solution.)
2. The error due to quadrature is hard to evaluate, with parallelised versions aggravating the error.
3. There is a truncation error due to the stopping rule when the exact maximum of the likelihood function is unknown.

Not mentioning the Monte Carlo error, of course, which should remain at the √n level.

“Nested Sampling is a special type of adaptive SMC algorithm, where weights are assigned in a suboptimal way.”

The above remark is somewhat obvious for a fixed sequence of likelihood levels and a set of particles at each (ring) level. moved by a Markov kernel with the right stationary target. Constrained to move within the ring, which may prove delicate in complex settings. Such a non-adaptive version is however not realistic and hence both the level sets and the stopping rule need be selected from the existing simulation, respectively as a quantile of the observed likelihood and as a failure to modify the evidence approximation, an adaptation that is a Catch 22! as we already found in the AMIS paper.  (AMIS stands for adaptive mixture importance sampling.) To escape the quandary, the authors use both an auxiliary variable (to avoid atoms) and two importance sampling sequences (as in AMIS). And only a single particle with non-zero incremental weight for the (upper level) target. As the full details are a bit fuzzy to me, I hope I can experiment with my (quarantined) students on the full implementation of the method.

“Such cases asides, the question whether SMC is preferable using the TA or NS approach is really one of whether it is preferable to sample (relatively) easy distributions subject to a constraint or to sample potentially difficult distributions.”

A question (why not regular SMC?) I was indeed considering until coming to the conclusion section but did not find it treated in the paper. There is little discussion on the computing requirements either, as it seems the method is more time-consuming than a regular nested sample. (On the personal side,  I appreciated very much their “special thanks to Christian Robert, whose many blog posts on NS helped influence this work, and played a large partin inspiring it.”)

Why do we draw parameters to draw from a marginal distribution that does not contain the parameters?

A revealing question on X validated of a simulation concept students (and others) have trouble gripping with. Namely using auxiliary variates to simulate from a marginal distribution, since these auxiliary variables are later dismissed and hence appear to them (students) of no use at all. Even after being exposed to the accept-reject algorithm. Or to multiple importance sampling. In the sense that a realisation of a random variable can be associated with a whole series of densities in an importance weight, all of them being valid (but some more equal than others!).

revisiting the balance heuristic

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.)

a new rule for adaptive importance sampling

Art Owen and Yi Zhou have arXived a short paper on the combination of importance sampling estimators. Which connects somehow with the talk about multiple estimators I gave at ESM last year in Helsinki. And our earlier AMIS combination. The paper however makes two important assumptions to reach optimal weighting, which is inversely proportional to the variance:

1. the estimators are uncorrelated if dependent;
2. the variance of the k-th estimator is of order a (negative) power of k.

The later is puzzling when considering a series of estimators, in that k appears to act as a sample size (as in AMIS), the power is usually unknown but also there is no reason for the power to be the same for all estimators. The authors propose to use ½ as the default, both because this is the standard Monte Carlo rate and because the loss in variance is then minimal, being 12% larger.

As an aside, Art Owen also wrote an invited discussion “the unreasonable effectiveness of Monte Carlo” of ” Probabilistic Integration: A Role in Statistical Computation?” by François-Xavier Briol, Chris  Oates, Mark Girolami (Warwick), Michael Osborne and Deni Sejdinovic, to appear in Statistical Science, discussion that contains a wealth of smart and enlightening remarks. Like the analogy between pseudo-random number generators [which work unreasonably well!] vs true random numbers and Bayesian numerical integration versus non-random functions. Or the role of advanced bootstrapping when assessing the variability of Monte Carlo estimates (citing a paper of his from 1992). Also pointing out at an intriguing MCMC paper by  Michael Lavine and Jim Hodges to appear in The American Statistician.