on using the data twice…

As I was writing my next column for CHANCE, I decided I will include a methodology box about “using the data twice”. Here is the draft. (The second part is reproduced verbatim from an earlier post on Error and Inference.)

Several aspects of the books covered in this CHANCE review [i.e., Bayesian ideas and data analysis, and Bayesian modeling using WinBUGS] face the problem of “using the data twice”. What does that mean? Nothing really precise, actually. The accusation of “using the data twice” found in the Bayesian literature can be thrown at most procedures exploiting the Bayesian machinery without actually being Bayesian, i.e.~which cannot be derived from the posterior distribution. For instance, the integrated likelihood approach in Murray Aitkin’s Statistical Inference avoids the difficulties related with improper priors π_i by first using the data x to construct (proper) posteriors π_i(θ_i|x) and then secondly using the data in a Bayes factor

as if the posteriors were priors. This obviously solves the improperty difficulty (see. e.g., The Bayesian Choice), but it creates a statistical procedure outside the Bayesian domain, hence requiring a separate validation since the usual properties of Bayesian procedures do not apply. Similarly, the whole empirical Bayes approach falls under this category, even though some empirical Bayes procedures are asymptotically convergent. The pseudo-marginal likelihood of Geisser and Eddy (1979), used in  Bayesian ideas and data analysis, is defined by

through the marginal posterior likelihoods. While it also allows for improper priors, it does use the same data in each term of the product and, again, it is not a Bayesian procedure.

Once again, from first principles, a Bayesian approach should use the data only once, namely when constructing the posterior distribution on every unknown component of the model(s).  Based on this all-encompassing posterior, all inferential aspects should be the consequences of a sequence of decision-theoretic steps in order to select optimal procedures. This is the ideal setting while, in practice,  relying on a sequence of posterior distributions is often necessary, each posterior being a consequence of earlier decisions, which makes it the result of a multiple (improper) use of the data… For instance, the process of Bayesian variable selection is on principle clean from the sin of “using the data twice”: one simply computes the posterior probability of each of the variable subsets and this is over. However, in a case involving many (many) variables, there are two difficulties: one is about building the prior distributions for all possible models, a task that needs to be automatised to some extent; another is about exploring the set of potential models. First, ressorting to projection priors as in the intrinsic solution of Pèrez and Berger (2002, Biometrika, a much valuable article!), while unavoidable and a “least worst” solution, means switching priors/posteriors based on earlier acceptances/rejections, i.e. on the data. Second, the path of models truly explored by a computational algorithm [which will be a minuscule subset of the set of all models] will depend on the models rejected so far, either when relying on a stepwise exploration or when using a random walk MCMC algorithm. Although this is not crystal clear (there is actually plenty of room for supporting the opposite view!), it could be argued that the data is thus used several times in this process…

Error and Inference [on wrong models]

In connection with my series of posts on the book Error and Inference, and my recent collation of those into an arXiv document, Deborah Mayo has started a series of informal seminars at the LSE on the philosophy of errors in statistics and the likelihood principle. and has also posted a long comment on my argument about only using wrong models. (The title is inspired from the Rolling Stones’ “You can’t always get what you want“, very cool!) The discussion about the need or not to take into account all possible models (which is the meaning of the “catchall hypothesis” I had missed while reading the book) shows my point was not clear. I obviously do not claim in the review that all possible models should be accounted for at once, this was on the opposite my understanding of Mayo’s criticism of the Bayesian approach (I thought the following sentence was clear enough: “According to Mayo, this alternative hypothesis should “include all possible rivals, including those not even though of” (p.37)”)! So I see the Bayesian approach as a way to put on the table a collection of reasonable (if all wrong) models and give to those models a posterior probability, with the purpose that improbable ones are eliminated. Therefore, I am in agreement with most of the comments in the post, esp. because this has little to do with Bayesian versus frequentist testing! Even rejecting the less likely models from a collection seems compatible with a Bayesian approach, model averaging is not always an appropriate solution, depending on the loss function!

Error and Inference [arXived]

Following my never-ending series of posts on the book Error and Inference, (edited) by Deborah Mayo and Ari Spanos (and kindly sent to me by Deborah), I decided to edit those posts into a (slightly) more coherent document, now posted on arXiv. And to submit it as a book review to Siam Review, even though I had not high expectations it fits the purpose of the journal: the review was rejected between the submission to arXiv and the publication of this post!

the cult of significance

“Statistical significance is not a scientific test. It is a philosophical, qualitative test. It asks “whether”. Existence, the question of whether, is interesting. But it is not scientific.” S. Ziliak and D. McCloskey, p.5

The book, written by economists Stephen Ziliak and Deirdre McCloskey, has a theme bound to attract Bayesians and all those puzzled by the absolute and automatised faith in significance tests. The main argument of the authors is indeed that an overwhelming majority of papers stop at rejecting variables (“coefficients”) on the sole and unsupported basis of non-significance at the 5% level. Hence the subtitle “How the standard error costs us jobs, justice, and lives“… This is an argument I completely agree with, however, the aggressive style of the book truly put me off! As with Error and Inference, which also addresses a non-Bayesian issue, I could have let the matter go, however I feel the book may in the end be counter-productive and thus endeavour to explain why through this review.  (I wrote the following review in batches, before and during my trip to Dublin, so the going is rather broken, I am afraid…)

Error and Inference [end]

(This is my sixth and last post on Error and Inference, being as previously a raw and naïve reaction born from a linear and sluggish reading of the book, rather than a deeper and more informed criticism with philosophical bearings. Read at your own risk.)

“‘It is refreshing to see Cox and Mayo give a hard-nosed statement of what scientific objectivity demands of an account of statistics, show how it relates to frequentist statistics, and contrast that with the notion of “objectivity” used by O-Bayesians.”—A. Spanos, p.326, Error and Inference, 2010

In order to conclude my pedestrian traverse of Error and Inference, I read the discussion by Aris Spanos of the second part of the seventh chapter by David Cox’s and Deborah Mayo’s, discussed in the previous post. (In the train to the half-marathon to be precise, which may have added a sharper edge to the way I read it!) The first point in the discussion is that the above paper is “a harmonious blend of the Fisherian and N-P perspectives to weave a coherent frequentist inductive reasoning anchored firmly on error probabilities”(p.316). The discussion by Spanos is very much a-critical of the paper, so I will not engage into a criticism of the non-criticism, but rather expose some thoughts of mine that came from reading this apology. (Remarks about Bayesian inference are limited to some piques like the above, which only reiterates those found earlier [and later: "the various examples Bayesians employ to make their case involve some kind of "rigging" of the statistical model", Aris Spanos, p.325; "The Bayesian epistemology literature is filled with shadows and illusions", Clark Glymour, p. 335] in the book.) [I must add I do like the mention of O-Bayesians, as I coined the O'Bayes motto for the objective Bayes bi-annual meetings from 2003 onwards! It also reminds me of the O-rings and of the lack of proper statistical decision-making in the Challenger tragedy...]

The “general frequentist principle for inductive reasoning” (p.319) at the core of Cox and Mayo’s paper is obviously the central role of the p-value in “providing (strong) evidence against the null H₀ (for a discrepancy from H₀)”. Once again, I fail to see it as the epitome of a working principle in that

1. it depends on the choice of a divergence d(z), which reduces the information brought by the data z;
2. it does not articulate the level for labeling nor the consequences of finding a low p-value;
3. it ignores the role of the alternative hypothesis.

Furthermore, Spanos’ discussion deals with “the fallacy of rejection” (pp.319-320) in a rather artificial (if common) way, namely by setting a buffer of discrepancy γ around the null hypothesis. While the choice of a maximal degree of precision sounds natural to me (in the sense that a given sample size should not allow for the discrimination between two arbitrary close values of the parameter), the fact that γ is in fine set by the data (so that the p-value is high) is fairly puzzling. If I understand correctly, the change from a p-value to a discrepancy γ is a fine device to make the “distance” from the null better understood, but it has an extremely limited range of application. If I do not understand correctly, the discrepancy γ is fixed by the statistician and then this sounds like an extreme form of prior selection.

There is at least one issue I do not understand in this part, namely the meaning of the severity evaluation probability

as the conditioning on the event seems impossible in a frequentist setting. This leads me to an idle and unrelated questioning as to whether there is a solution to

as this would be the ultimate discrepancy. Or whether this does not make any sense… because of the ambiguous role of z₀, which needs somehow to be integrated out. (Otherwise, d can be chosen so that the probability is 1.)

“If one renounces the likelihood, the stopping rule, and the coherence principles, marginalizes the use of prior information as largely untrustworthy, and seek procedures with `good’ error probabilistic properties (whatever that means), what is left to render the inference Bayesian, apart from a belief (misguided in my view) that the only way to provide an evidential account of inference is to attach probabilities to hypotheses?”—A. Spanos, p.326, Error and Inference, 2010

The role of conditioning ancillary statistics is emphasized both in the paper and the discussion. This conditioning clearly reduces variability, however there is no reservation about the arbitrariness of such ancillary statistics. And the fact that conditioning any further would lead to conditioning upon the whole data, i.e. to a Bayesian solution. I also noted a curious lack of proper logical reasoning in the argument that, when

using the conditional ancillary distribution is enough, since, while “any departure from f(z|s) implies that the overall model is false” (p.322), but not the reverse. Hence, a poor choice of s may fail to detect a departure. (Besides the fact that  fixed-dimension sufficient statistics do not exist outside exponential families.) Similarly, Spanos expands about the case of a minimal sufficient statistic that is independent from a maximal ancillary statistic, but such cases are quite rare and limited to exponential families [in the iid case]. Still in the conditioning category, he also supports Mayo’s argument against the likelihood principle being a consequence of the sufficiency and weak conditionality principles. A point I discussed in a previous post. However, he does not provide further evidence against Birnbaum’s result, arguing rather in favour of a conditional frequentist inference I have nothing to complain about. (I fail to perceive the appeal of the Welch uniform example in terms of the likelihood principle.)

In an overall conclusion, let me repeat and restate that this series of posts about Error and Inference is far from pretending at bringing a Bayesian reply to the philosophical arguments raised in the volume. The primary goal being of “taking some crucial steps towards legitimating the philosophy of frequentist statistics” (p.328), I should not feel overly concerned. It is only when the debate veered towards a comparison with the Bayesian approach [often too often of the "holier than thou" brand] that I felt allowed to put in my twopennies worth… I do hope I may crystallise this set of notes into a more constructed review of the book, if time allows, although I am pessimistic at the chances of getting it published given our current difficulties with the critical review of Murray Aitkin’s  Statistical Inference. However, as a coincidence, we got back last weekend an encouraging reply from Statistics and Risk Modelling, prompting us towards a revision and the prospect of a reply by Murray.