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).

ABC random forests for Bayesian parameter inference

Before leaving Helsinki, we arXived [from the Air France lounge!] the paper Jean-Michel presented on Monday at ABCruise in Helsinki. This paper summarises the experiments Louis conducted over the past months to assess the great performances of a random forest regression approach to ABC parameter inference. Thus validating in this experimental sense the use of this new approach to conducting ABC for Bayesian inference by random forests. (And not ABC model choice as in the Bioinformatics paper with Pierre Pudlo and others.)

I think the major incentives in exploiting the (still mysterious) tool of random forests [against more traditional ABC approaches like Fearnhead and Prangle (2012) on summary selection] are that (i) forests do not require a preliminary selection of the summary statistics, since an arbitrary number of summaries can be used as input for the random forest, even when including a large number of useless white noise variables; (b) there is no longer a tolerance level involved in the process, since the many trees in the random forest define a natural if rudimentary distance that corresponds to being or not being in the same leaf as the observed vector of summary statistics η(y); (c) the size of the reference table simulated from the prior (predictive) distribution does not need to be as large as for in usual ABC settings and hence this approach leads to significant gains in computing time since the production of the reference table usually is the costly part! To the point that deriving a different forest for each univariate transform of interest is truly a minor drag in the overall computing cost of the approach.

An intriguing point we uncovered through Louis’ experiments is that an unusual version of the variance estimator is preferable to the standard estimator: we indeed exposed better estimation performances when using a weighted version of the out-of-bag residuals (which are computed as the differences between the simulated value of the parameter transforms and their expectation obtained by removing the random trees involving this simulated value). Another intriguing feature [to me] is that the regression weights as proposed by Meinshausen (2006) are obtained as an average of the inverse of the number of terms in the leaf of interest. When estimating the posterior expectation of a transform h(θ) given the observed η(y), this summary statistic η(y) ends up in a given leaf for each tree in the forest and all that matters for computing the weight is the number of points from the reference table ending up in this very leaf. I do find this difficult to explain when confronting the case when many simulated points are in the leaf against the case when a single simulated point makes the leaf. This single point ends up being much more influential that all the points in the other situation… While being an outlier of sorts against the prior simulation. But now that I think more about it (after an expensive Lapin Kulta beer in the Helsinki airport while waiting for a change of tire on our airplane!), it somewhat makes sense that rare simulations that agree with the data should be weighted much more than values that stem from the prior simulations and hence do not translate much of an information brought by the observation. (If this sounds murky, blame the beer.) What I found great about this new approach is that it produces a non-parametric evaluation of the cdf of the quantity of interest h(θ) at no calibration cost or hardly any. (An R package is in the making, to be added to the existing R functions of abcrf we developed for the ABC model choice paper.)

abcfr 0.9-3

In conjunction with our reliable ABC model choice via random forest paper, about to be resubmitted to Bioinformatics, we have contributed an R package called abcrf that produces a most likely model and its posterior probability out of an ABC reference table. In conjunction with the realisation that we could devise an approximation to the (ABC) posterior probability using a secondary random forest. “We” meaning Jean-Michel Marin and Pierre Pudlo, as I only acted as a beta tester!

The package abcrf consists of three functions:

• abcrf, which constructs a random forest from a reference table and returns an object of class `abc-rf’;
• plot.abcrf, which gives both variable importance plot of a model choice abc-rf object and the projection of the reference table on the LDA axes;
• predict.abcrf, which predict the model for new data and evaluate the posterior probability of the MAP.

An illustration from the manual:

data(snp)
data(snp.obs)
mc.rf <- abcrf(snp[1:1e3, 1], snp[1:1e3, -1])
predict(mc.rf, snp[1:1e3, -1], snp.obs)