## robust Bayesian synthetic likelihood

Posted in Statistics with tags , , , , , , , , , , , , , on May 16, 2019 by xi'an

David Frazier (Monash University) and Chris Drovandi (QUT) have recently come up with a robustness study of Bayesian synthetic likelihood that somehow mirrors our own work with David. In a sense, Bayesian synthetic likelihood is definitely misspecified from the start in assuming a Normal distribution on the summary statistics. When the data generating process is misspecified, even were the Normal distribution the “true” model or an appropriately converging pseudo-likelihood, the simulation based evaluation of the first two moments of the Normal is biased. Of course, for a choice of a summary statistic with limited information, the model can still be weakly compatible with the data in that there exists a pseudo-true value of the parameter θ⁰ for which the synthetic mean μ(θ⁰) is the mean of the statistics. (Sorry if this explanation of mine sounds unclear!) Or rather the Monte Carlo estimate of μ(θ⁰) coincidences with that mean.The same Normal toy example as in our paper leads to very poor performances in the MCMC exploration of the (unsympathetic) synthetic target. The robustification of the approach as proposed in the paper is to bring in an extra parameter to correct for the bias in the mean, using an additional Laplace prior on the bias to aim at sparsity. Or the same for the variance matrix towards inflating it. This over-parameterisation of the model obviously avoids the MCMC to get stuck (when implementing a random walk Metropolis with the target as a scale).

## asymptotics of synthetic likelihood [a reply from the authors]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on March 19, 2019 by xi'an

[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.

## asymptotics of synthetic likelihood

Posted in pictures, Statistics, Travel with tags , , , , , , , , , , on March 11, 2019 by xi'an

David Nott, Chris Drovandi and Robert Kohn just arXived a paper on a comparison between ABC and synthetic likelihood, which is both interesting and timely given that synthetic likelihood seems to be lacking behind in terms of theoretical evaluation. I am however as puzzled by the results therein as I was by the earlier paper by Price et al. on the same topic. Maybe due to the Cambodia jetlag, which is where and when I read the paper.

My puzzlement, thus, comes from the difficulty in comparing both approaches on a strictly common ground. The paper first establishes convergence and asymptotic normality for synthetic likelihood, based on the 2003 MCMC paper of Chernozukov and Hong [which I never studied in details but that appears like the MCMC reference in the econometrics literature]. The results are similar to recent ABC convergence results, unsurprisingly when assuming a CLT on the summary statistic vector. One additional dimension of the paper is to consider convergence for a misspecified covariance matrix in the synthetic likelihood [and it will come back with a revenge]. And asymptotic normality of the synthetic score function. Which is obviously unavailable in intractable models.

The first point I have difficulty with is how the computing time required for approximating mean and variance in the synthetic likelihood, by Monte Carlo means, is not accounted for in the comparison between ABC and synthetic likelihood versions. Remember that ABC only requires one (or at most two) pseudo-samples per parameter simulation. The latter requires M, which is later constrained to increase to infinity with the sample size. Simulations that are usually the costliest in the algorithms. If ABC were to use M simulated samples as well, since it already relies on a kernel, it could as well construct [at least on principle] a similar estimator of the [summary statistic] density. Or else produce M times more pairs (parameter x pseudo-sample). The authors pointed out (once this post out) that they do account for the factor M when computing the effective sample size (before Lemma 4, page 12), but I still miss why the ESS converging to N=MN/M when M goes to infinity is such a positive feature.

Another point deals with the use of multiple approximate posteriors in the comparison. Since the approximations differ, it is unclear that convergence to a given approximation is all that should matter, if the approximation is less efficient [when compared with the original and out-of-reach posterior distribution]. Especially for a finite sample size n. This chasm in the targets becomes more evident when the authors discuss the use of a constrained synthetic likelihood covariance matrix towards requiring less pseudo-samples, i.e. lower values of M, because of a smaller number of parameters to estimate. This should be balanced against the loss in concentration of the synthetic approximation, as exemplified by the realistic examples in the paper. (It is also hard to see why M could be not of order √n for Monte Carlo reasons.)

The last section in the paper is revolving around diverse issues for misspecified models, from wrong covariance matrix to wrong generating model. As we just submitted a paper on ABC for misspecified models, I will not engage into a debate on this point but find the proposed strategy that goes through an approximation of the log-likelihood surface by a Gaussian process and a derivation of the covariance matrix of the score function apparently greedy in both calibration and computing. And not so clearly validated when the generating model is misspecified.

## Bayesian gan [gan style]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , on June 26, 2018 by xi'an

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.]

## 1500 nuances of gan [gan gan style]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on February 16, 2018 by xi'an

I recently realised that there is a currently very popular trend in machine learning called GAN [for generative adversarial networks] that strongly connects with ABC, at least in that it relies mostly on the availability of a generative model, i.e., a probability model that can be generated as in . For instance, there was a GANs tutorial at NIPS 2016 by Ian Goodfellow and many talks on the topic at recent NIPS, the 1500 in the title referring to the citations of the GAN paper by Goodfellow et al. (2014). (The name adversarial comes from opposing true model to generative model in the inference. )

If you remember Jeffreys‘s famous pique about classical tests as being based on improbable events that did not happen, GAN, like ABC,  is sort of the opposite in that it generates events until the one that was observed happens. More precisely, by generating pseudo-samples and switching parameters until these samples get as confused as possible between the data generating (“true”) distribution and the generative one. (In its original incarnation, GAN is indeed an optimisation scheme in .) A basic presentation of GAN is that it constructs a function D(x,ϕ) that represents the probability that x came from the true model p versus the generative model, ϕ being the parameter of a neural network trained to this effect, aimed at minimising in ϕ a two-term objective function

where the first expectation is taken under the true model and the second one under the generative model.

“The discriminator tries to best distinguish samples away from the generator. The generator tries to produce samples that are indistinguishable by the discriminator.” Edward

One ABC perception of this technique is that the confusion rate

is a form of distance between the data and the generative model. Which expectation can be approximated by repeated simulations from this generative model. Which suggests an extension from the optimisation approach to a ABCyesian version by selecting the smallest distances across a range of θ‘s simulated from the prior.

This notion relates to solution using classification tools as density ratio estimation, connecting for instance to Gutmann and Hyvärinen (2012). And ultimately with Geyer’s 1992 normalising constant estimator.

Another link between ABC and networks also came out during that trip. Proposed by Bishop (1994), mixture density networks (MDN) are mixture representations of the posterior [with component parameters functions of the data] trained on the prior predictive through a neural network. These MDNs can be trained on the ABC learning table [based on a specific if redundant choice of summary statistics] and used as substitutes to the posterior distribution, which brings an interesting alternative to Simon Wood’s synthetic likelihood. In a paper I missed Papamakarios and Murray suggest replacing regular ABC with this version…