Incoherent inference

“The probability of the nested special case must be less than or equal to the probability of the general model within which the special case is nested. Any statistic that assigns greater probability to the special case is incoherent. An example of incoherence is shown for the ABC method.” Alan Templeton, PNAS, 2010

Alan Templeton just published an article in PNAS about “coherent and incoherent inference” (with applications to phylogeography and human evolution). While his (misguided) arguments are mostly those found in an earlier paper of his’ and discussed in this post as well as in the defence of model based inference twenty-two of us published in Molecular Ecology a few months ago, the paper contains a more general critical perspective on Bayesian model comparison, aligning argument after argument about the incoherence of the Bayesian approach (and not of ABC, as presented there). The notion of coherence is borrowed from the 1991 (Bayesian) paper of Lavine and Schervish on Bayes factors, which shows that Bayes factors may be nonmonotonous in the alternative hypothesis (but also that posterior probabilities aren’t!). Templeton’s first argument proceeds from the quote above, namely that larger models should have larger probabilities or else this violates logic and coherence! The author presents the reader with a Venn diagram to explain why a larger set should have a larger measure. Obviously, he does not account for the fact that in model choice, different models induce different parameters spaces and that those spaces are endowed with orthogonal measures, especially if the spaces are of different dimensions. In the larger space, . (This point is not even touching the issue of defining “the” probability over the collection of models that Templeton seems to take for granted but that does not make sense outside a Bayesian framework.) Talking therefore of nested models having a smaller probability than the encompassing model or of “partially overlapping models” does not make sense from a measure theoretic (hence mathematical) perspective. (The fifty-one occurences of coherent/incoherent in the paper do not bring additional weight to the argument!)

“Approximate Bayesian computation (ABC) is presented as allowing statistical comparisons among models. ABC assigns posterior probabilities to a finite set of simulated a priori models.” Alan Templeton, PNAS, 2010

An issue common to all recent criticisms by Templeton is the misleading or misled confusion between the ABC method and the resulting Bayesian inference. For instance, Templeton distinguishes between the incoherence in the ABC model choice procedure from the incoherence in the Bayes factor, when ABC is used as a computational device to approximate the Bayes factor. There is therefore no inferential aspect linked with ABC,  per se, it is simply a numerical tool to approximate Bayesian procedures and, with enough computer power, the approximation can get as precise as one wishes. In this paper, Templeton also reiterates the earlier criticism that marginal likelihoods are not comparable across models, because they “are not adjusted for the dimensionality of the data or the models” (sic!). This point is missing the whole purpose of using marginal likelihoods, namely that they account for the dimensionality of the parameter by providing a natural Ockham’s razor penalising the larger model without requiring to specify a penalty term. (If necessary, BIC is so successful! provides an approximation to this penalty, as well as the alternate DIC.) The second criticism of ABC (i.e. of the Bayesian approach) is that model choice requires a collection of models and cannot decide outside this collection. This is indeed the purpose of a Bayesian model choice and studies like Berger and Sellke (1987, JASA) have shown the difficulty of reasoning within a single model. Furthermore, Templeton advocates the use of a likelihood ratio test, which necessarily implies using two models. Another Venn diagram also explains why Bayes formula when used for model choice is “mathematically and logically incorrect” because marginal likelihoods cannot be added up when models “overlap”: according to him, “there can be no universal denominator, because a simple sum always violates the constraints of logic when logically overlapping models are tested“. Once more, this simply shows a poor understanding of the probabilistic modelling involved in model choice.

“The central equation of ABC is inherently incoherent for three separate reasons, two of which are applicable in every case that deals with overlapping hypotheses.” Alan Templeton, PNAS, 2010

This argument relies on the representation of the “ABC equation” (sic!)

where is the observed summary statistic, is “the vector of expected (simulated) summary statistics under model ” and “ is a goodness-of-fit measure“. Templeton states that this “fundamental equation is mathematically incorrect in every instance (..) of overlap.” This representation of the ABC approximation is again misleading or misled in that the simulation algorithm ABC produces an approximation to a posterior sample from . The resulting approximation to the marginal likelihood under model is a regular Monte Carlo step that replaces an integral with a weighted sum, not a “goodness-of-fit measure.”  The subsequent argument  of Templeton’s about the goodness-of-fit measures being “not adjusted for the dimensionality of the data” (re-sic!) and the resulting incoherence is therefore void of substance. The following argument repeats an earlier misunderstanding with the probabilistic model involved in Bayesian model choice: the reasoning that, if

“the constraints of logic are violated [and] the prior probabilities used in the very first step of their Bayesian analysis are incoherent“, does not assimilate the issue of measures over mutually exclusive spaces.

“ABC is used for parameter estimation in addition to hypothesis testing and another source of incoherence is suggested from the internal discrepancy between the posterior probabilities generated by ABC and the parameter estimates found by ABC.” Alan Templeton, PNAS, 2010

The point corresponding to the above quote is that, while the posterior probability that (model ) is much higher than the posterior probability of the opposite (model ), the Bayes estimate of under model is “significantly different from zero“. Again, this reflects both a misunderstanding of the probability model, namely that is impossible [has measure zero] under model , and a confusion between confidence intervals (that are model specific) and posterior probabilities (that work across models). The concluding message that “ABC is a deeply flawed Bayesian procedure in which ignorance overwhelms data to create massive incoherence” is thus unsubstantiated.

“Incoherent methods, such as ABC, Bayes factor, or any simulation approach that treats all hypotheses as mutually exclusive, should never be used with logically overlapping hypotheses.” Alan Templeton, PNAS, 2010

In conclusion, I am quite surprised at this controversial piece of work being published in PNAS, as the mathematical and statistical arguments of Professor Templeton should have been assessed by referees who are mathematicians and statisticians, in which case they would have spotted the obvious inconsistencies!

Defence of model-based inference

A tribune—to which I contributed—about the virtues of statistical inference in phylogeography  just appeared in Molecular Ecology. (The whole paper seems to be available on line as I can access it.) It has been written by 22 (!) contributors in response to Templeton’s recent criticism of ABC (and his defence of Nested Clade Analysis) in the same journal. My contribution to the paper is mostly based on the arguments posted here last March, namely that the paper was confusing ABC (which is a computational method) with Bayesian statistics. The paper as a whole goes beyond a “Bayesian defence” since not all authors are Bayesian. It supports a statistics based approach to phyleogeography, as reported in the abstract

Recent papers have promoted the view that model-based methods in general, and those based on Approximate Bayesian Computation (ABC) in particular, are flawed in a number of ways, and are therefore inappropriate for the analysis of phylogeographic data. These papers further argue that Nested Clade Phylogeographic Analysis (NCPA) offers the best approach in statistical phylogeography. In order to remove the confusion and misconceptions introduced by these papers, we justify and explain the reasoning behind model-based inference. We argue that ABC is a statistically valid approach, alongside other computational statistical techniques that have been successfully used to infer parameters and compare models in population genetics. We also examine the NCPA method and highlight numerous deficiencies, either when used with single or multiple loci. We further show that the ages of clades are carelessly used to infer ages of demographic events, that these ages are estimated under a simple model of panmixia and population stationarity but are then used under different and unspecified models to test hypotheses, a usage the invalidates these testing procedures. We conclude by encouraging researchers to study and use model-based inference in population genetics.

This will most presumably fail to end the debate between the proponents and the opponents of model-based inference in phylogenics and elsewhere, but the point was worth making…