asymptotics of synthetic likelihood [a reply from the authors]

[Here is a reply from David, Chris, and Robert on my earlier comments, highlighting some points I had missed or misunderstood.]

Dear Christian

Thanks for your interest in our synthetic likelihood paper and the thoughtful comments you wrote about it on your blog.  We’d like to respond to the comments to avoid some misconceptions.

Your first claim is that we don’t account for the differing number of simulation draws required for each parameter proposal in ABC and synthetic likelihood.  This doesn’t seem correct, see the discussion below Lemma 4 at the bottom of page 12.  The comparison between methods is on the basis of effective sample size per model simulation.

As you say, in the comparison of ABC and synthetic likelihood, we consider the ABC tolerance \epsilon and the number of simulations per likelihood estimate M in synthetic likelihood as functions of n.  Then for tuning parameter choices that result in the same uncertainty quantification asymptotically (and the same asymptotically as the true posterior given the summary statistic) we can look at the effective sample size per model simulation.  Your objection here seems to be that even though uncertainty quantification is similar for large n, for a finite n the uncertainty quantification may differ.  This is true, but similar arguments can be directed at almost any asymptotic analysis, so this doesn’t seem a serious objection to us at least.  We don’t find it surprising that the strong synthetic likelihood assumptions, when accurate, give you something extra in terms of computational efficiency.

We think mixing up the synthetic likelihood/ABC comparison with the comparison between correctly specified and misspecified covariance in Bayesian synthetic likelihood is a bit unfortunate, since these situations are quite different.  The first involves correct uncertainty quantification asymptotically for both methods.  Only a very committed reader who looked at our paper in detail would understand what you say here.  The question we are asking with the misspecified covariance is the following.  If the usual Bayesian synthetic likelihood analysis is too much for our computational budget, can something still be done to quantify uncertainty?  We think the answer is yes, and with the misspecified covariance we can reduce the computational requirements by an order of magnitude, but with an appropriate cost statistically speaking.  The analyses with misspecified covariance give valid frequentist confidence regions asymptotically, so this may still be useful if it is all that can be done.  The examples as you say show something of the nature of the trade-off involved.

We aren’t quite sure what you mean when you are puzzled about why we can avoid having M to be O(√n).  Note that because of the way the summary statistics satisfy a central limit theorem, elements of the covariance matrix of S are already O(1/n), and so, for example, in estimating μ(θ) as an average of M simulations for S, the elements of the covariance matrix of the estimator of μ(θ) are O(1/(Mn)).  Similar remarks apply to estimation of Σ(θ).  I’m not sure whether that gets to the heart of what you are asking here or not.

In our email discussion you mention the fact that if M increases with n, then the computational burden of a single likelihood approximation and hence generating a single parameter sample also increases with n.  This is true, but unavoidable if you want exact uncertainty quantification asymptotically, and M can be allowed to increase with n at any rate.  With a fixed M there will be some approximation error, which is often small in practice.  The situation with vanilla ABC methods will be even worse, in terms of the number of proposals required to generate a single accepted sample, in the case where exact uncertainty quantification is desired asymptotically.  As shown in Li and Fearnhead (2018), if regression adjustment is used with ABC and you can find a good proposal in their sense, one can avoid this.  For vanilla ABC, if the focus is on point estimation and exact uncertainty quantification is not required, the situation is better.  Of course as you show in your nice ABC paper for misspecified models jointly with David Frazier and Juidth Rousseau recently the choice of whether to use regression adjustment can be subtle in the case of misspecification.

In our previous paper Price, Drovandi, Lee and Nott (2018) (which you also reviewed on this blog) we observed that if the summary statistics are exactly normal, then you can sample from the summary statistic posterior exactly with finite M in the synthetic likelihood by using pseudo-marginal ideas together with an unbiased estimate of a normal density due to Ghurye and Olkin (1962).  When S satisfies a central limit theorem so that S is increasingly close to normal as n gets large, we conjecture that it is possible to get exact uncertainty quantification asymptotically with fixed M if we use the Ghurye and Olkin estimator, but we have no proof of that yet (if it is true at all).

Thanks again for being interested enough in the paper to comment, much appreciated.

David, Chris, Robert.

the paper where you are a node

Sophie Donnet pointed out to me this arXived paper by Tianxi Li, Elizaveta Levina, and Ji Zhu, on a network resampling strategy for X validation, where I appear as a datapoint rather than as a [direct] citation! Which reminded me of the “where you are the hero” gamebooks with which my kids briefly played, before computer games took over. The model selection method is illustrated on a dataset made of X citations [reduced to 706 authors]  in all papers published between 2003 and 2012 in the Annals of Statistics, Biometrika, JASA, and JRSS Series B. With the outcome being the determination of a number of communities, 20, which the authors labelled as they wanted, based on 10 authors with the largest number of citations in the category. As it happens, I appear in the list, within the “mixed (causality + theory + Bayesian)” category (!), along with Jamie Robbins, Paul Fearnhead, Gilles Blanchard, Zhiqiang Tan, Stijn Vansteelandt, Nancy Reid, Jae Kwang Kim, Tyler VanderWeele, and Scott Sisson, which is somewhat mind-boggling in that I am pretty sure I never quoted six of these authors [although I find it hilarious that Jamie appears in the category, given that we almost got into a car crash together, at one of the Valencià meetings!].

Max Ent at Max Plank

let the evidence speak [book review]

This book by Alan Jessop, professor at the Durham University Business School,  aims at presenting Bayesian ideas and methods towards decision making “without formula because they are not necessary; the ability to add and multiply is all that is needed.” The trick is in using a Bayes grid, in other words a two by two table. (There are a few formulas that survived the slaughter, see e.g. on p. 91 the formula for the entropy. Contained in the chapter on information that I find definitely unclear.) When leaving the 2×2 world, things become more complicated and the construction of a prior belief as a probability density gets heroic without the availability of maths formulas. The first part of the paper is about Likelihood, albeit not the likelihood function, despite having the general rule that (p.73)

belief is proportional to base rate x likelihood

which is the book‘s version of Bayes’ (base?!) theorem. It then goes on to discuss the less structure nature of prior (or prior beliefs) against likelihood by describing Tony O’Hagan’s way of scaling experts’ beliefs in terms of a Beta distribution. And mentioning Jaynes’ maximum entropy prior without a single formula. What is hard to fathom from the text is how can one derive the likelihood outside surveys. (Using the illustration of 1963 Oswald’s murder by Ruby in the likelihood chapter does not particularly help!) A bit of nitpicking at this stage: the sentence

“The ancient Greeks, and before them the Chinese and the Aztecs…”

is historically incorrect since, while the Chinese empire dates back before the Greek dark ages, the Aztecs only rule Mexico from the 14th century (AD) until the Spaniard invasion. While most of the book sticks with unidimensional parameters, it also discusses more complex structures, for which it relies on Monte Carlo, although the description is rather cryptic (use your spreadsheet!, p.133). The book at this stage turns into a more story-telling mode, by considering for instance the Federalist papers analysis by Mosteller and Wallace. The reader can only follow the process of assessing a document authorship for a single word, as multidimensional cases (for either data or parameters) are out of reach. The same comment applies to the ecology, archeology, and psychology chapters that follow. The intermediary chapter on the “grossly misleading” [Court wording] of the statistical evidence in the Sally Clark prosecution is more accessible in that (again) it relies on a single number. Returning to the ban of Bayes rule in British courts:

In the light of the strong criticism by this court in the 1990s of using Bayes theorem before the jury in cases where there was no reliable statistical evidence, the practice of using a Bayesian approach and likelihood ratios to formulate opinions placed before a jury without that process being disclosed and debated in court is contrary to principles of open justice.

the discussion found in the book is quite moderate and inclusive, in that a Bayesian analysis helps in gathering evidence about a case, but may be misunderstood or misused at the [non-Bayesian] decision level.

In conclusion, Let the Evidence Speak is an interesting introduction to Bayesian thinking, through a simplifying device, the Bayes grid, which seems to come from management, with a large number of examples, if not necessarily all realistic and some side-stories. I doubt this exposure can produce expert practitioners, but it makes for an worthwhile awakening for someone “likely to have read this book because [one] had heard of Bayes but were uncertain what is was” (p.222). With commendable caution and warnings along the way.

visual effects

As advertised and re-discussed by Dan Simpson on the Statistical Modeling, &tc. blog he shares with Andrew and a few others, the paper Visualization in Bayesian workflow he wrote with Jonah Gabry, Aki Vehtari, Michael Betancourt and Andrew Gelman was one of three discussed at the RSS conference in Cardiff, last week month, as a Read Paper for Series A. I had stored the paper when it came out towards reading and discussing it, but as often this good intention led to no concrete ending. [Except concrete as in concrete shoes…] Hence a few notes rather than a discussion in Series B A.

Exploratory data analysis goes beyond just plotting the data, which should sound reasonable to all modeling readers.

Fake data [not fake news!] can be almost [more!] as valuable as real data for building your model, oh yes!, this is the message I am always trying to convey to my first year students, when arguing about the connection between models and simulation, as well as a defense of ABC methods. And more globally of the very idea of statistical modelling. While indeed “Bayesian models with proper priors are generative models”, I am not particularly fan of using the prior predictive [or the evidence] to assess the prior as it may end up in a classification of more or less all but terrible priors, meaning that all give very little weight to neighbourhoods of high likelihood values. Still, in a discussion of a TAS paper by Seaman et al. on the role of prior, Kaniav Kamary and I produced prior assessments that were similar to the comparison illustrated in Figure 4. (And this makes me wondering which point we missed in this discussion, according to Dan.)  Unhappy am I with the weakly informative prior illustration (and concept) as the amount of fudging and calibrating to move from the immensely vague choice of N(0,100) to the fairly tight choice of N(0,1) or N(1,1) is not provided. The paper reads like these priors were the obvious and first choice of the authors. I completely agree with the warning that “the utility of the the prior predictive distribution to evaluate the model does not extend to utility in selecting between models”.

MCMC diagnostics, beyond trace plots, yes again, but this recommendation sounds a wee bit outdated. (As our 1998 reviewww!) Figure 5(b) links different parameters of the model with lines, which does not clearly relate to a better understanding of convergence. Figure 5(a) does not tell much either since the green (divergent) dots stand within the black dots, at least in the projected 2D plot (and how can one reach beyond 2D?) Feels like I need to rtfm..!

“Posterior predictive checks are vital for model evaluation”, to wit that I find Figure 6 much more to my liking and closer to my practice. There could have been a reference to Ratmann et al. for ABC where graphical measures of discrepancy were used in conjunction with ABC output as direct tools for model assessment and comparison. Essentially predicting a zero error with the ABC posterior predictive. And of course “posterior predictive checking makes use of the data twice, once for the fitting and once for the checking.” Which means one should either resort to loo solutions (as mentioned in the paper) or call for calibration of the double-use by re-simulating pseudo-datasets from the posterior predictive. I find the suggestion that “it is a good idea to choose statistics that are orthogonal to the model parameters” somewhat antiquated, in that this sounds like rephrasing the primeval call to ancillary statistics for model assessment (Kiefer, 1975), while pretty hard to implement in modern complex models.

ISBA 18 tidbits

Among a continuous sequence of appealing sessions at this ISBA 2018 meeting [says a member of the scientific committee!], I happened to attend two talks [with a wee bit of overlap] by Sid Chib in two consecutive sessions, because his co-author Ana Simoni (CREST) was unfortunately sick. Their work was about models defined by a collection of moment conditions, as often happens in econometrics, developed in a recent JASA paper by Chib, Shin, and Simoni (2017). With an extension about moving to defining conditional expectations by use of a functional basis. The main approach relies on exponentially tilted empirical likelihoods, which reminded me of the empirical likelihood [BCel] implementation we ran with Kerrie Mengersen and Pierre Pudlo a few years ago. As a substitute to ABC. This problematic made me wonder on how much Bayesian the estimating equation concept is, as it should somewhat involve a nonparametric prior under the moment constraints.

Note that Sid’s [talks and] papers are disconnected from ABC, as everything comes in closed form, apart from the empirical likelihood derivation, as we actually found in our own work!, but this could become a substitute model for ABC uses. For instance, identifying the parameter θ of the model by identifying equations. Would that impose too much input from the modeller? I figure I came with this notion mostly because of the emphasis on proxy models the previous day at ABC in ‘burgh! Another connected item of interest in the work is the possibility of accounting for misspecification of these moment conditions by introducing a vector of errors with a spike & slab distribution, although I am not sure this is 100% necessary without getting further into the paper(s) [blame conference pressure on my time].

Another highlight was attending a fantastic poster session Monday night on computational methods except I would have needed four more hours to get through every and all posters. This new version of ISBA has split the posters between two sites (great) and themes (not so great!), while I would have preferred more sites covering all themes over all nights, to lower the noise (still bearable this year) and to increase the possibility to check all posters of interest in a particular theme…

Mentioning as well a great talk by Dan Roy about assessing deep learning performances by what he calls non-vacuous error bounds. Namely, through PAC-Bayesian bounds. One major comment of his was about deep learning models being much more non-parametric (number of parameters rising with number of observations) than parametric models, meaning that generative adversarial constructs as the one I discussed a few days ago may face a fundamental difficulty as models are taken at face value there.

On closed-form solutions, a closed-form Bayes factor for component selection in mixture models by Fũqene, Steel and Rossell that resemble the Savage-Dickey version, without the measure theoretic difficulties. But with non-local priors. And closed-form conjugate priors for the probit regression model, using unified skew-normal priors, as exhibited by Daniele Durante. Which are product of Normal cdfs and pdfs, and which allow for closed form marginal likelihoods and marginal posteriors as well. (The approach is not exactly conjugate as the prior and the posterior are not in the same family.)

And on the final session I attended, there were two talks on scalable MCMC, one on coresets, which will require some time and effort to assimilate, by Trevor Campbell and Tamara Broderick, and another one using Poisson subsampling. By Matias Quiroz and co-authors. Which did not completely convinced me (but this was the end of a long day…)

All in all, this has been a great edition of the ISBA meetings, if quite intense due to a non-stop schedule, with a very efficient organisation that made parallel sessions manageable and poster sessions back to a reasonable scale [although I did not once manage to cross the street to the other session]. Being in unreasonably sunny Edinburgh helped a lot obviously! I am a wee bit disappointed that no one else follows my call to wear a kilt, but I had low expectations to start with… And too bad I missed the Ironman 70.3 Edinburgh by one day!

Bayesian gan [gan style]

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.]