likelihood-free approximate Gibbs sampling

“Low-dimensional regression-based models are constructed for each of these conditional distributions using synthetic (simulated) parameter value and summary statistic pairs, which then permit approximate Gibbs update steps (…) synthetic datasets are not generated during each sampler iteration, thereby providing efficiencies for expensive simulator models, and only require sufficient synthetic datasets to adequately construct the full conditional models (…) Construction of the approximate conditional distributions can exploit known structures of the high-dimensional posterior, where available, to considerably reduce computational overheads”

Guilherme Souza Rodrigues, David Nott, and Scott Sisson have just arXived a paper on approximate Gibbs sampling. Since this comes a few days after we posted our own version, here are some of the differences I could spot in the paper:

1. Further references to earlier occurrences of Gibbs versions of ABC, esp. in cases when the likelihood function factorises into components and allows for summaries with lower dimensions. And even to ESP.
2. More an ABC version of Gibbs sampling that a Gibbs version of ABC in that approximations to the conditionals are first constructed and then used with no further corrections.
3. Inherently related to regression post-processing à la Beaumont et al.  (2002) in that the regression model is the start to designing an approximate full conditional, conditional on the “other” parameters and on the overall summary statistic. The construction of the approximation is far from automated. And may involve neural networks or other machine learning estimates.
4. As a consequence of the above, a preliminary ABC step to design the collection of approximate full conditionals using a single and all-purpose multidimensional summary statistic.
5. Once the approximations constructed, no further pseudo-data is generated.
6. Drawing from the approximate full conditionals is done exactly, possibly via a bootstrapped version.
7. Handling a highly complex g-and-k dynamic model with 13,140 unknown parameters, requiring a ten days simulation.

“In certain circumstances it can be seen that the likelihood-free approximate Gibbs sampler will exactly target the true partial posterior (…) In this case, then Algorithms 2 and 3 will be exact.”

Convergence and coherence are handled in the paper by setting the algorithm(s) as noisy Monte Carlo versions, à la Alquier et al., although the issue of incompatibility between the full conditionals is acknowledged, with the main reference being the finite state space analysis of Chen and Ip (2015). It thus remains unclear whether or not the Gibbs samplers that are implemented there do converge and if they do what is the significance of the stationary distribution.

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.

ABC²DE

A recent arXival on a new version of ABC based on kernel estimators (but one could argue that all ABC versions are based on kernel estimators, one way or another.) In this ABC-CDE version, Izbicki,  Lee and Pospisilz [from CMU, hence the picture!] argue that past attempts failed to exploit the full advantages of kernel methods, including the 2016 ABCDE method (from Edinburgh) briefly covered on this blog. (As an aside, CDE stands for conditional density estimation.) They also criticise these attempts at selecting summary statistics and hence failing in sufficiency, which seems a non-issue to me, as already discussed numerous times on the ‘Og. One point of particular interest in the long list of drawbacks found in the paper is the inability to compare several estimates of the posterior density, since this is not directly ingrained in the Bayesian construct. Unless one moves to higher ground by calling for Bayesian non-parametrics within the ABC algorithm, a perspective which I am not aware has been pursued so far…

The selling points of ABC-CDE are that (a) the true focus is on estimating a conditional density at the observable x⁰ rather than everywhere. Hence, rejecting simulations from the reference table if the pseudo-observations are too far from x⁰ (which implies using a relevant distance and/or choosing adequate summary statistics). And then creating a conditional density estimator from this subsample (which makes me wonder at a double use of the data).

The specific density estimation approach adopted for this is called FlexCode and relates to an earlier if recent paper from Izbicki and Lee I did not read. As in many other density estimation approaches, they use an orthonormal basis (including wavelets) in low dimension to estimate the marginal of the posterior for one or a few components of the parameter θ. And noticing that the posterior marginal is a weighted average of the terms in the basis, where the weights are the posterior expectations of the functions themselves. All fine! The next step is to compare [posterior] estimators through an integrated squared error loss that does not integrate the prior or posterior and does not tell much about the quality of the approximation for Bayesian inference in my opinion. It is furthermore approximated by  a doubly integrated [over parameter and pseudo-observation] squared error loss, using the ABC(ε) sample from the prior predictive. And the approximation error only depends on the regularity of the error, that is the difference between posterior and approximated posterior. Which strikes me as odd, since the Monte Carlo error should take over but does not appear at all. I am thus unclear as to whether or not the convergence results are that relevant. (A difficulty with this paper is the strong dependence on the earlier one as it keeps referencing one version or another of FlexCode. Without reading the original one, I spotted a mention made of the use of random forests for selecting summary statistics of interest, without detailing the difference with our own ABC random forest papers (for both model selection and estimation). For instance, the remark that “nuisance statistics do not affect the performance of FlexCode-RF much” reproduces what we observed with ABC-RF.

The long experiment section always relates to the most standard rejection ABC algorithm, without accounting for the many alternatives produced in the literature (like Li and Fearnhead, 2018. that uses Beaumont et al’s 2002 scheme, along with importance sampling improvements, or ours). In the case of real cosmological data, used twice, I am uncertain of the comparison as I presume the truth is unknown. Furthermore, from having worked on similar data a dozen years ago, it is unclear why ABC is necessary in such context (although I remember us running a test about ABC in the Paris astrophysics institute once).

random forests [reading group]

Here are the slides I prepared (and recycled) over the weekend for the reading group on machine-learning that recently started in Warwick. Where I am for two consecutive weeks.

machine learning-based approach to likelihood-free inference

At ABC’ory last week, Kyle Cranmer gave an extended talk on estimating the likelihood ratio by classification tools. Connected with a 2015 arXival. The idea is that the likelihood ratio is invariant by a transform s(.) that is monotonic with the likelihood ratio itself. It took me a few minutes (after the talk) to understand what this meant. Because it is a transform that actually depends on the parameter values in the denominator and the numerator of the ratio. For instance the ratio itself is a proper transform in the sense that the likelihood ratio based on the distribution of the likelihood ratio under both parameter values is the same as the original likelihood ratio. Or the (naïve Bayes) probability version of the likelihood ratio. Which reminds me of the invariance in Fearnhead and Prangle (2012) of the Bayes estimate given x and of the Bayes estimate given the Bayes estimate. I also feel there is a connection with Geyer’s logistic regression estimate of normalising constants mentioned several times on the ‘Og. (The paper mentions in the conclusion the connection with this problem.)

Now, back to the paper (which I read the night after the talk to get a global perspective on the approach), the ratio is of course unknown and the implementation therein is to estimate it by a classification method. Estimating thus the probability for a given x to be from one versus the other distribution. Once this estimate is produced, its distributions under both values of the parameter can be estimated by density estimation, hence an estimated likelihood ratio be produced. With better prospects since this is a one-dimensional quantity. An objection to this derivation is that it intrinsically depends on the pair of parameters θ¹ and θ² used therein. Changing to another pair requires a new ratio, new simulations, and new density estimations. When moving to a continuous collection of parameter values, in a classical setting, the likelihood ratio involves two maxima, which can be formally represented in (3.3) as a maximum over a likelihood ratio based on the estimated densities of likelihood ratios, except that each evaluation of this ratio seems to require another simulation. (Which makes the comparison with ABC more complex than presented in the paper [p.18], since ABC major computational hurdle lies in the production of the reference table and to a lesser degree of the local regression, both items that can be recycled for any new dataset.) A smoothing step is then to include the pair of parameters θ¹ and θ² as further inputs of the classifier.  There still remains the computational burden of simulating enough values of s(x) towards estimating its density for every new value of θ¹ and θ². And while the projection from x to s(x) does effectively reduce the dimension of the problem to one, the method still aims at estimating with some degree of precision the density of x, so cannot escape the curse of dimensionality. The sleight of hand resides in the classification step, since it is equivalent to estimating the likelihood ratio. I thus fail to understand how and why a poor classifier can then lead to a good approximations of the likelihood ratio “obtained by calibrating s(x)” (p.16). Where calibrating means estimating the density.