*(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

**I**n 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: "t*he various examples Bayesians employ to make their case involve some kind of "rigging" of the statistical model*", Aris Spanos, p.325; "T*he 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...]

**T**he “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 H_{0} (for a discrepancy from H_{0})”. Once again, I fail to see it as the epitome of a working principle in that

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

**T**here 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 z_{0}, 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

**T**he 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.)

**I**n 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.