# 60 yrs of Cox’s (1958) weighing machine, & links to binge-read the Likelihood Principle

December 27, 2017
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(This article was originally published at Statistics – Error Statistics Philosophy, and syndicated at StatsBlogs.)

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2018 will mark 60 years since the famous chestnut from Sir David Cox (1958). The example  “is now usually called the ‘weighing machine example,’ which draws attention to the need for conditioning, at least in certain types of problems” (Reid 1992, p. 582). When I describe it, you’ll find it hard to believe many regard it as causing an earthquake in statistical foundations, unless you’re already steeped in these matters. A simple version: If half the time I reported my weight from a scale that’s always right, and half the time use a scale that gets it right with probability .5, would you say I’m right with probability ¾? Well, maybe. But suppose you knew that this measurement was made with the scale that’s right with probability .5? The overall error probability is scarcely relevant for giving the warrant of the particular measurement, knowing which scale was used. So what’s the earthquake? First a bit more on the chestnut. Here’s an excerpt from Cox and Mayo (2010, 295-8):

p.296p.298

Binging the Likelihood Principle. The earthquake grows from the fact that it has long been thought that the (WCP) entails the (strong) Likelihood Principle (LP), based on a famous argument by Allan Birnbaum (1962). But the LP renders error probabilities irrelevant to parametric inference once the data are known. J. Savage calls Birnbaum’s argument “a landmark in statistics” (see [i]). I give a disproof of Birnbaum’s argument (via counterexample) in Mayo (2010), but later saw the need for a deeper argument which I give in Mayo (2014) in Statistical Science.[ii] (There, among other subtleties, the WCP is put as a logical equivalence as intended.)

“It was the adoption of an unqualified equivalence formulation of conditionality, and related concepts, which led, in my 1962 paper, to the monster of the likelihood axiom,” (Birnbaum 1975, 263).

If you’re keen to binge on Birnbaum’s brainbuster, perhaps to break holiday/winter break doldrums, I’ve pasted most of the early historical sources below. The argument is simple; showing what’s wrong with it took a long time. You can also find quite a lot on the LP searching this blog (including posts by readers); it was a main topic for the first few years of this blog. You might start with a summary post (based on slides) here, or an intermediate paper Mayo (2013) I presented at the JSM. In it I ask:

Does it matter? On the face of it, current day uses of sampling theory statistics do not seem in need of going back 50 years to tackle a foundational argument. This may be so, but only if it is correct to assume that the Birnbaum argument must be flawed somewhere.(Mayo 2013, p.441)

It is.

[i] Savage on Birnbaum: “This paper is a landmark in statistics. . . . I, myself, like other Bayesian statisticians, have been convinced of the truth of the likelihood principle for a long time. Its consequences for statistics are very great. . . . [T]his paper is really momentous in the history of statistics. It would be hard to point to even a handful of comparable events. …once the likelihood principle is widely recognized, people will not long stop at that halfway house but will go forward and accept the implications of personalistic probability for statistics” (Savage 1962, 307-308).
The argument purports to follow from principles frequentist error statisticians accept.

[ii] The link includes comments on my paper by Bjornstad, Dawid, Evans, Fraser, Hannig, and Martin and Liu, and my rejoinder.

Birnbaum Papers:

• Birnbaum, A. (1962), “On the Foundations of Statistical Inference“, Journal of the American Statistical Association 57(298), 269-306.
• Savage, L. J., Barnard, G., Cornfield, J., Bross, I, Box, G., Good, I., Lindley, D., Clunies-Ross, C., Pratt, J., Levene, H., Goldman, T., Dempster, A., Kempthorne, O, and Birnbaum, A. (1962). “Discussion on Birnbaum’s On the Foundations of Statistical Inference”, Journal of the American Statistical Association 57(298), 307-326.
• Birnbaum, A (1970). Statistical Methods in Scientific Inference (letter to the editor). Nature 225, 1033.
• Birnbaum, A (1972), “More on Concepts of Statistical Evidence“, Journal of the American Statistical Association, 67(340), 858-861.

Durbin:

Evans, Fraser, and Monette:

Kalbfleisch:

References for the Blogpost:

Birnbaum, A. (1962), “On the Foundations of Statistical Inference“, Journal of the American Statistical Association 57(298), 269-306.

Birnbaum, A. (1975). Comments on Paper by J. D. Kalbfleisch. Biometrika, 62 (2), 262–264.

Cox, D. R. (1958), “Some problems connected with statistical inference“, The Annals of Mathematical Statistics, 29, 357-372.

Cox D. R. and Mayo. D. G. (2010). “Objectivity and Conditionality in Frequentist Inference” in Error and Inference: Recent Exchanges on Experimental Reasoning, Reliability and the Objectivity and Rationality of Science (D Mayo and A. Spanos eds.), Cambridge: Cambridge University Press: 276-304.

Mayo, D. G. (2010). “An Error in the Argument from Conditionality and Sufficiency to the Likelihood Principle” in Error and Inference: Recent Exchanges on Experimental Reasoning, Reliability and the Objectivity and Rationality of Science (D Mayo and A. Spanos eds.), Cambridge: Cambridge University Press: 305-14.

Mayo, D. G. (2013) “Presented Version: On the Birnbaum Argument for the Strong Likelihood Principle”, in JSM Proceedings, Section on Bayesian Statistical Science. Alexandria, VA: American Statistical Association: 440-453.

Mayo, D. G. (2014). Mayo paper: “On the Birnbaum Argument for the Strong Likelihood Principle,” Paper with discussion and Mayo rejoinder: Statistical Science 29(2) pp. 227-239, 261-266.

Reid, N. (1992). Introduction to Fraser (1966) structural probability and a generalization. In Breakthroughs in Statistics (S. Kotz and N. L. Johnson, eds.) 579–586. Springer Series in Statistics. Springer, New York.

Savage, L. J. (1962). Discussion on a paper by A. Birnbaum [On the foundations of statistical inference]. Journal of the American Statistical Association, 57, 307– 308.

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