Comment and Reply 439

way, so that we have not shared the common experience which would doubtless have

brought us even closer together.

References

Barnard, G.A. (1980). Pivotal inference and the Bayesian controversy. Proceedings ofthe First Vulencia Conference on Bayesian Statistics - Trab. Estadistica 31, 295-318.

Barnard, G.A. (1985). Pivotal inference. In: N.L. Johnson and S. Kotz, Eds., Encyclopedia of Statistical Sciences. John Wiley, New York.

A REPLY TO THE COMMENT BY G.A. BARNARD

I.J. GOOD

Department of Statistics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. U.S. A.

It is not my purpose now to comment on the lectures given at this conference. But

I would like to thank everyone who attended, to compliment the speakers on their

fine presentations, and to thank Dennis Lindley for his Introduction.

But George Barnard’s Comment does I think invite a reply. I agree with him that

our basic philosophies are close, but I cannot say how close until I have studied his

theory of pivotal inference with more care and understanding. I also agree with him

that, in common with many others, our early backgrounds in applications might

well have affected our philosophies. (During World War II we met in London and

I told Barnard that in Bletchley we were using weights of evidence sequentially to

discriminate between two hypotheses. He told me that he was using essentially the

same method for quality control in the Ministry of Supply. He has forgotten the

meeting.) But I had read some of the writings of F.P. Ramsay, J.M. Keynes and

H. Jeffreys before I was seriously involved with applications and was much in-

fluenced by those writers.

Barnard (1987) has reminded us that Fisher eventually regarded the fiducial argu-

ment as almost exactly of the sort used by Bayes in his billiard-table argument. I

published criticisms of the fiducial argument some time ago (Good, 1965, pp. 81-83;

1971) but Barnard’s reminder has provoked me to think again about Bayes’s

billiard-table argument. Fortunately Barnard had the initiative to get Bayes’s paper

(Bayes, 1763) reprinted in Biometrika for the great convenience of the statistical

profession.

440 and Reply

I am intending to publish my new thoughts on Bayes’s article in the Journal of Stastistical Computation and Simulation under the title “Bayes’s red billiard ball

is also a herring, and why Bayes withheld publication”, so I shall not say much

about it here. Note only that Bayes’s argument cannot be used to arrive at posterior

probabilities (from a binomial sample) unless something like a ‘Bayes Postulate’ is

assumed. I believe that the same is true of the fiducial argument so in in this respect

at least the arguments of Bayes and Fisher seem to have something in common. But

Fisher implicitly claimed that posterior probabilities could sometimes be evaluated

without assuming priors, whereas Bayes claimed that the appropriate way to express

ignorance about the number of successes in a binomial sample was to assume that

it had a uniform distribution, discrete of course.

In Barnard’s comment he raises the matter of adhockery. I do not regard ad-

hockery as necessarily bad, in fact I said in Good (1965, p. 56) that “We make no

mockery of honest adhockery”. Also I believe there is a kind of measure of degrees

of adhockery (Good, 1983a, and the first index of 1983b where there is a mistake

in sign), but I do not claim to have said the last word on that topic!

References

Barnard, G.A. (1987). F.A. Fisher - a true Bayesian? Internat. Statist. Rev. 55, 183-189.

Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. Reprinted in Biometrika 45 (1958), 296-311, with an introduction by G.A. Barnard, and an appendix by Richard Price.

Good, I.J. (1965). The Estimation of Probabilities: An Essay on Modern Bayesian Methods. MIT Press,

Cambridge, MA.

Good, I.J. (1971). Response to comments by G.A. Barnard. In: V.P. Godambe and D.A. Sprott, Eds.,

Foundations of Statistical Inference. Holt, Rinehart and Winston, Toronto, 138-140.

Good, I.J. (1983a). A measure of adhockery. Cl45 in J. Statist. Cornput. Simul. 16, 314.

Good, I.J. (1983b). Good Thinking: The Foundations of Probability and its Applications. University of

Minnesota Press, Minneapolis, MN.