## 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.)

## politics coming [too close to] statistics [or the reverse]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , on May 9, 2020 by xi'an

On 30 April, David Spiegelhalter wrote an opinion column in The Guardian, Coronavirus deaths: how does Britain compare with other countries?, where he pointed out the difficulty, even “for a bean-counting statistician to count deaths”, as the reported figures are undercounts, and stated that “many feel that excess deaths give a truer picture of the impact of an epidemic“. Which, on the side, I indeed believe is a more objective material, as also reported by INSEE and INED in France.

“…my cold, statistical approach is to wait until the end of the year, and the years after that, when we can count the excess deaths. Until then, this grim contest won’t produce any league tables we can rely on.” D. Spiegelhalter

My understanding of the tribune is that the quick accumulation of raw numbers, even for deaths, and their use in the comparison of procedures and countries is not helping in understanding the impacts of policies and actions-reactions from a week ago. Starting with the delays in reporting death certificates, as again illustrated by the ten day lag in the INSEE reports. And accounting for covariates such as population density, economic and health indicators. (The graph below for instance relies on deaths so far attributed to COVID-19 rather than on excess deaths, while these attributions depend on the country policy and its official statistics capacities.)

“Polite request to PM and others: please stop using my Guardian article to claim we cannot make any international comparisons yet. I refer only to detailed league tables—of course we should now use other countries to try and learn why our numbers are high.” D. Spiegelhalter

However, when on 6 May Boris Johnson used this Guardian article during prime minister’s questions in the UK Parliement, to defuse a question from the Labour leader, Keir Starmer, David Spiegelhalter reacted with the above tweet, which is indeed that even with poor and undercounted data the total number of cases is much worse than predicted by the earlier models and deadlier than in neighbouring countries. Anyway, three other fellow statisticians, Phil Brown, Jim Smith (Warwick), and Henry Wynn, also reacted to David’s tribune by complaining at the lack of statistical modelling behind it and the fatalistic message it carries, advocating for model based decision-making, which would be fine if the data was not so unreliable… or if the proposed models were equipped with uncertainty bumpers accounting for misspecification and erroneous data.

## rare ABC [webinar impressions]

Posted in Books, Statistics, Travel, University life with tags , , , , , , , on April 28, 2020 by xi'an

A second occurrence of the One World ABC seminar by Ivis Kerama, and Richard Everitt (Warwick U), on their on-going pape with and Tom Thorne, Rare Event ABC-SMC², which is not about rare event simulation but truly about ABC improvement. Building upon a previous paper by Prangle et al. (2018). And also connected with Dennis’ talk a fortnight ago in that it exploits an autoencoder representation of the simulated outcome being H(u,θ). It also reminded me of an earlier talk by Nicolas Chopin.

This approach avoids using summary statistics (but relies on a particular distance) and implements a biased sampling of the u’s to produce outcomes more suited to the observation(s). Almost sounds like a fiducial ABC! Their stopping rule for decreasing the tolerance is to spot an increase in the variance of the likelihood estimates. As the method requires many data generations for a single θ, it only applies in certain settings. The ABC approximation is indeed used as an estimation of likelihood ratio (which makes sense for SMC² but is biased because of ABC). I got slightly confused during Richard’s talk by his using the term of unbiased estimator of the likelihood before I realised he was talking of the ABC posterior. Thanks to both speakers, looking forward the talk by Umberto Picchini in a fortnight (on a joint paper with Richard).

## back to the Bernoulli factory

Posted in Books, Statistics, University life with tags , , , on April 7, 2020 by xi'an

“The results show that the proposed algorithm is asymptotically optimal for the mentioned subclass of functions, in the sense that for any other fast algorithm E[N] grows at least as fast with p.”

Murray Pollock (Warwick U. for a wee more days!) pointed out to me this paper of Luis Mendo on a Bernoulli factory algorithm that estimates functions [of p] that can be expressed as power series [of p]. Essentially functions f(p) such that f(0)=0 and f(1)=1. The big difference with earlier algorithms, as far as I can tell, is that the approach involves a randomised stopping rule that involves, on top of the unlimited sequence of Bernoulli B(p) variates a second sequence of Uniform variates, which sounds to me like a change of paradigm, given the much higher degree of freedom brought by Uniform variates (as opposed to Bernoulli variates with an unknown value of p). Although there exists a non-randomised version in the paper. The proposed algorithm is as follows, using a sequence of d’s issued from the power series coefficients:

1. Set i=1.
2. Take one input X[i].
3. Produce U[i] uniform on (0,1). Let V[i]=1 if U[i]<d[i] and V[i]=0 otherwise.
If V[i] or X[i] are equal to 1, output X[i] and finish.
Else increase i and go back to step2.

As the author mentions, this happens to be a particular case of the reverse-time martingale approach of Łatuszynski, Kosmidis, Papaspiliopoulos and Roberts (Warwick connection as well!). With an average number of steps equal to f(p)/p, surprisingly simple, and somewhat of an optimal rate. While the functions f(p) are somewhat restricted, this is nice work

## ABC World seminar

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , on April 4, 2020 by xi'an

With most of the World being more or less confined at home, conferences cancelled one after the other, including ABC in Grenoble!, we are launching a fortnightly webinar on approximation Bayesian computation, methods, and inference. The idea is to gather members and disseminate results and innovation during these coming weeks and months under lock-down. And hopefully after!

At this point, the interface will be Blackboard Collaborate, run from Edinburgh by Michael Gutmann, for which neither registration nor software is required. Before each talk, a guest link will be mailed to the mailing list. Please register here to join the list.

The seminar is planned on Thursdays at either 9am or more likely 11:30 am UK (+1GMT) time, as we are still debating the best schedule to reach as many populated time zones as possible!, and the first speakers are

 09.04.2020 Dennis Prangle Distilling importance sampling 23.04.2020 Ivis Kerama and Richard Everitt Rare event SMC² 07.05.2020 Umberto Picchini Stratified sampling and bootstrapping for ABC