Archive for tolerance

adaptive ABC tolerance

Posted in Books, Statistics, University life with tags , , , , , , , , , on June 2, 2020 by xi'an

“There are three common approaches for selecting the tolerance sequence (…) [they] can lead to inefficient sampling”

Umberto Simola, Jessi Cisewski-Kehe, Michael Gutmann and Jukka Corander recently arXived a paper entitled Adaptive Approximate Bayesian Computation Tolerance Selection. I appreciate that they start from our ABC-PMC paper, i.e., Beaumont et al. (2009) [although the representation that the ABC tolerances are fixed in advance is somewhat incorrect in that we used in our codes quantiles of the distances to set our tolerances.] This is also the approach advocated for the initialisation step by the current paper.  Although remaining a wee bit vague. Subsequent steps are based on the proximity between the resulting approximations to the ABC posteriors, more exactly with a quantile derived from the maximum of the ratio between two estimated successive ABC posteriors. Mimicking the Accept-Reject step if always one step too late.  The iteration stops when the ratio is almost one, possibly missing the target due to Monte Carlo variability. (Recall that the “optimal” tolerance is not zero for a finite sample size.)

“…the decrease in the acceptance rate is mitigated by the improvement in the proposed particles.”

A problem is that it depends on the form of the approximation and requires non-parametric hence imprecise steps. Maybe variational encoders could help. Interesting approach by Sugiyama et al. (2012), of which I knew nothing, the core idea being that the ratio of two densities is also the solution to minimising a distance between the numerator density and a variable function times the bottom density. However since only the maximum of the ratio is needed, a more focused approach could be devised. Rather than first approximating the ratio and second maximising the estimated ratio. Maybe the solution of Goffinet et al. (1992) on estimating an accept-reject constant could work.

A further comment is that the estimated density is not properly normalised, which lessens the Accept-Reject analogy since the optimum may well stand above one. And thus stop “too soon”. (Incidentally, the paper contains the mixture example of Sisson et al. (2007), for which our own graphs were strongly criticised during our Biometrika submission!)

ABC with Gibbs steps

Posted in Statistics with tags , , , , , , , , , , , , , , , , , on June 3, 2019 by xi'an

With Grégoire Clarté, Robin Ryder and Julien Stoehr, all from Paris-Dauphine, we have just arXived a paper on the specifics of ABC-Gibbs, which is a version of ABC where the generic ABC accept-reject step is replaced by a sequence of n conditional ABC accept-reject steps, each aiming at an ABC version of a conditional distribution extracted from the joint and intractable target. Hence an ABC version of the standard Gibbs sampler. What makes it so special is that each conditional can (and should) be conditioning on a different statistic in order to decrease the dimension of this statistic, ideally down to the dimension of the corresponding component of the parameter. This successfully bypasses the curse of dimensionality but immediately meets with two difficulties. The first one is that the resulting sequence of conditionals is not coherent, since it is not a Gibbs sampler on the ABC target. The conditionals are thus incompatible and therefore convergence of the associated Markov chain becomes an issue. We produce sufficient conditions for the Gibbs sampler to converge to a stationary distribution using incompatible conditionals. The second problem is then that, provided it exists, the limiting and also intractable distribution does not enjoy a Bayesian interpretation, hence may fail to be justified from an inferential viewpoint. We however succeed in producing a version of ABC-Gibbs in a hierarchical model where the limiting distribution can be explicited and even better can be weighted towards recovering the original target. (At least with limiting zero tolerance.)

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.

approximate likelihood perspective on ABC

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , on December 20, 2018 by xi'an

George Karabatsos and Fabrizio Leisen have recently published in Statistics Surveys a fairly complete survey on ABC methods [which earlier arXival I had missed]. Listing within an extensive bibliography of 20 pages some twenty-plus earlier reviews on ABC (with further ones in applied domains)!

“(…) any ABC method (algorithm) can be categorized as either (1) rejection-, (2) kernel-, and (3) coupled ABC; and (4) synthetic-, (5) empirical- and (6) bootstrap-likelihood methods; and can be combined with classical MC or VI algorithms [and] all 22 reviews of ABC methods have covered rejection and kernel ABC methods, but only three covered synthetic likelihood, one reviewed the empirical likelihood, and none have reviewed coupled ABC and bootstrap likelihood methods.”

The motivation for using approximate likelihood methods is provided by the examples of g-and-k distributions, although the likelihood can be efficiently derived by numerical means, as shown by Pierre Jacob‘s winference package, of mixed effect linear models, although a completion by the mixed effects themselves is available for Gibbs sampling as in Zeger and Karim (1991), and of the hidden Potts model, which we covered by pre-processing in our 2015 paper with Matt Moores, Chris Drovandi, Kerrie Mengersen. The paper produces a general representation of the approximate likelihood that covers the algorithms listed above as through the table below (where t(.) denotes the summary statistic):

The table looks a wee bit challenging simply because the review includes the synthetic likelihood approach of Wood (2010), which figured preeminently in the 2012 Read Paper discussion but opens the door to all kinds of approximations of the likelihood function, including variational Bayes and non-parametric versions. After a description of the above versions (including a rather ignored coupled version) and the special issue of ABC model choice,  the authors expand on the difficulties with running ABC, from multiple tuning issues, to the genuine curse of dimensionality in the parameter (with unnecessary remarks on low-dimension sufficient statistics since they are almost surely inexistent in most realistic settings), to the mis-specified case (on which we are currently working with David Frazier and Judith Rousseau). To conclude, an worthwhile update on ABC and on the side a funny typo from the reference list!

Li, W. and Fearnhead, P. (2018, in press). On the asymptotic efficiency
of approximate Bayesian computation estimators. Biometrika na na-na.

Approximate Bayesian computation via sufficient dimension reduction

Posted in Statistics, University life with tags , , , , , on August 26, 2016 by xi'an

“One of our contribution comes from the mathematical analysis of the consequence of conditioning the parameters of interest on consistent statistics and intrinsically inconsistent statistics”

Xiaolong Zhong and Malay Ghosh have just arXived an ABC paper focussing on the convergence of the method. And on the use of sufficient dimension reduction techniques for the construction of summary statistics. I had not heard of this approach before so read the paper with interest. I however regret that the paper does not link with the recent consistency results of Liu and Fearnhead and of Daniel Frazier, Gael Martin, Judith Rousseau and myself. When conditioning upon the MLE [or the posterior mean] as the summary statistic, Theorem 1 states that the Bernstein-von Mises theorem holds, missing a limit in the tolerance ε. And apparently missing conditions on the speed of convergence of this tolerance to zero although the conditioning event involves the true value of the parameter. This makes me wonder at the relevance of the result. The part about partial posteriors and the characterisation of limiting posterior distributions stats with the natural remark that the mean of the summary statistic must identify the whole parameter θ to achieve consistency, a point central to our 2014 JRSS B paper. The authors suggest using a support vector machine to derive the summary statistics, an idea already exploited by Heiko Strathmann et al.. There is no consistency result of relevance for ABC in that second and final part, which ends up rather abruptly. Overall, while the paper contributes to the current reflection on the convergence properties of ABC, the lack of scaling of the tolerance ε calls for further investigations.