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I. J. Good: An Appreciation

Module by: Stephen Fienberg. E-mail the authorEdited By: David Banks, Frederick Moody, eric smith

(This module helps introduce The Good Book: Thirty Years of Comments, Conjectures and Conclusions, by I.J. Good. The book is available for purchase from the Rice University Press Store. You can also visit the Rice University Press web site.)

I. J. Good—An Appreciation

I was privileged to attend a ninetieth birthday celebration in December 2006 for one of the major figures of twentieth-century statistics, and one of my personal heroes—I.J. Good, known to most of us as Jack. My interactions with Jack, both personal and intellectual, go back over forty years and I am greatly in his debt. It is a pleasure to be able to comment, albeit briefly, on some of his contributions to the literature that have influenced me the most.

December would have also marked the ninetieth birthday of another major figure and hero of mine, Fred Mosteller, who passed away after an extended illness in July 2006. It was while working on my initial research project under Fred's guidance as a graduate student at Harvard that I first encountered Jack's work. I stumbled across a paperback copy of The Scientist Speculates (Good, 1962), and in it I found a short piece on assessing probability assessors, by Bruno de Finetti (1962). At the time I was trying to polish up a memorandum (Tukey, 1965) on the topic initiated by some notes from John Tukey. Both the memo and the de Finetti piece were to exert a strong influence on my later collaboration with Morrie DeGroot on this topic—after we both had a conversation about it with Jack at the first Valencia meeting in 1979. (It was at Valencia that I first heard Jack describe the role of “fuzzy” priors at the top level of the Bayesian hierarchy!) But as important as that note by de Finetti was, it was in fact the rest of the volume that was so fascinating. And of course most of the entries were written by Jack on almost every topic one could imagine. “Who was this man?” I thought. I was soon to find out.

My next encounter with Jack's work came shortly afterwards. It was his then newly-published book on Estimation of Probabilities (Good, 1965) and his related 1956 article (Good, 1956) on small frequencies in contingency tables. They had a profound influence on my work at the time and reinforced my emerging commitment to the Bayesian perspective. I used ideas from this work in my own dissertation research and continued to go back to Jack's “little book” repeatedly in subsequent years, especially as I came to fully appreciate his explication of the notion of hierarchical models and mixtures of priors, not just for contingency table problems. It was also here that I learned about the importance of mixtures of Dirichlet distributions for contingency table problems, a topic Jack returned to repeatedly in subsequent years (e.g., see (Good, 1976) ). My original copy of the book still sits on my office shelf, somewhat dog-eared, with many penciled notes and question marks in the margins.

I believe I first met Jack at a professional meeting just after I received my Ph.D. (perhaps in Pittsburgh, although he is likely to remember better than I!), and I believe he spoke on Bode's law (Good, 1969). It was only after going to the University of Chicago where I began to work on log-linear models for multi-dimensional contingency tables that I discovered Jack's remarkable 1963 Annals paper (Good, 1963) on the use of entropy and marginals to generate log-linear models. Along with key papers by Birch, Darroch, Goodman, and Plackett, and of course Yvonne Bishop's thesis, that paper served as the foundation of my own work on the topic. Over the years I have had many occasions to refer others to it, when they “rediscovered” Jack's insights and approach.

By this time I had begun to assemble a collection of reprints and other copies of Jack's papers. I recall writing to him and asking for a copy of one and receiving a remarkable note in return. the key paragraph went something like:

According to my files I sent you a reprint of the requested paper on December 17, 1970. I am enclosing a copy of my short publication list, and suggest you request a different paper.

Of course, by then Jack's short publication list was longer than virtually any of the papers, and I dutifully made a different selection!

While working on my book with Yvonne Bishop and Paul Holland (1975), I had repeated occasions to go back to Jack's papers. One that I studied in particular, not yet mentioned, was his 1953 paper on the frequency of frequencies (Good, 1954) which suggests a method for summarizing large contingency tables by tabulating the frequency of the frequencies and then fitting distributions to it; this was also the “species” problem as in “How many words did Shakespeare know?” When it came time to choose a publisher for our book, an influential factor in our choice of MIT Press was the fact that it was the publisher of Jack's 1965 book; that was company we were pleased to keep. Both his book and ours were mainstays in the MIT Press catalogue for decades. As committed as Jack has been to the subjective Bayesian or personal probability perspective, his research papers show a remarkably catholic perspective on approaches to inference, and he has written repeatedly and at length about the Bayes/non-Bayes-compromise, e.g., (Good and Crook, 1974), as a matter of methodological approach rather than simply one of practice. Fred Mosteller too shared this pluralistic perspective, although he was never a committed subjectivist. This is part of the intellectual tradition to which I would like to be linked, being myself a true subjectivist philosophically, but practically using lots of maximum likelihood tools and indeed anything else that I can justify heuristically if not philosophically. How many kinds of Bayesians are there? According to Jack: 46,656 varieties (1971), and I suspect that several of these correspond to Jack wearing different hats, including his alter ego, K. Caj Doog. See also (Good, 1983b).

The reprinting of a collection of Jack's papers in 1983 (Good, 1983a) gave me a bound version of some of his work to keep close at hand and I have repeatedly referred to it, especially on matters on philosophy and on Bayesian and contingency table history.

Jack's interest in statistical fallacies and paradoxes is long-standing, including his 1968 encyclopedia article (Good, 1968) and his comment on Colin Blyth's article on Simpson's paradox (Good, 1972), so it was not a surprise when he wrote a lengthy piece on the topic, with new results and generalizations (Good and Mittal, 1985). It is a wonderful resource and it has been used not only by me but also by students with whom I have worked.

A little over four years ago I set out to answer the question: When did Bayesian inference become “Bayesian,” i.e., when did the term supplant “inverse probability”? I turned of course to several things that Jack had written, including the historical account in his 1965 book on The Estimation of Probabilities and I shared an early draft of my findings with Jack. This prompted a couple of e-mail messages, a long letter, a mailing with reprints or Xerox copies of several papers describing his wartime work at Bletchley Park with Turing and later during the 1950s, several of which he had sent me before (I guess he had stopped keeping track of the people to whom he had sent copies!), and a couple of telephone conversations. In the process I learned much both about the evolution of Bayesian methods and ideas, and Jack's role in what I now refer to as the “neo-Bayesian revival” of the 1950s, a term he coined and a movement in which he played an important part. My paper (2006) would not have been the same without him. And in case you are curious, he was not the first to write about “Bayesian inference,” even though he has many other firsts to his credit!

For me, virtually everything that Jack has written or opined about makes for “Good" reading and serious reflection—except perhaps for his limericks. I'm very pleased to be able to contribute to the present collection.

References

Bishop, Y.M.M., S.E. Fienberg, and P.W. Holland (1975). Discrete Multivariate Analysis: Theory and Practice. MIT Press. Reprinted (2007) Springer-Verlag.

de Finetti, B. (1962). “Does It Make Sense To Speak Of `Good Probability Appraisers'?”

DeGroot, M.H. and S.E. Fienberg (1982). “Assessing Probability Assessors: Calibration and Refinement,” In S.S. Gupta and J.O. Berger, eds., Statistical Decision Theory and Related Topics III, Vol. 1, Academic Press, 291–314.

Fienberg, S.E. (2006). “When Did Baysian Inference Become “Bayesian”?” Bayesian Analysis, 1, 1–40.

Good, I.J. (1954). “The Population Frequencies of Species and the Estimation of Population Parameters,” Biometrika, 40, 237–264.

Good, I.J. (1956). “On the Estimation of Small Frequencies in Contingency Tables,” J. Roy. Statist. Soc., Series B, 18. 113–124.

Good, I.J., ed. (with the help of A.J. Mayne and John Maynard Smith) (1962). The Scientists Speculates: An Anthology of Partly-Baked Ideas, London, Heinmann, 1962 (Basic Books, New York, 1963; Paperback, Capricorn Books, New York, 1965.)

Good, I.J. (1963). “Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables,” Ann. Math. Statist., 34, 911–934.

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

Good, I.J. (1968). “Fallacies, Statistical,” International Enc. Social Sciences, Macmillian and Free Press, 5, 292–301.

Good, I.J. (1969). “A Subjective Evaluation of Bode's Law and an Objective Test for Approximate Numerical Rationality (with discussion),” J. Amer. Statist. Assoc., 64, 23–66.

Good, I.J. (1971) “46656 varieties of Bayesians.” Letter in American Statistician, 25: 62– 63. Reprinted in Good Thinking, (Good 1983a), pp. 20–21.

Good, I.J. (1972). “Comments on Colin Blyth's two papers: On Simpson's Paradox and the Sure-thing Principle and Some Probability Paradoxes in Choice From Among Random alternatives,” J. Amer. Statist. Assoc., 67, 374–375.

Good, I.J. and J.F. Crook (1974). “The Bayes/Non-Bayes Compromise and the Multinomial Distribution,” J. Amer. Statist. Assoc., 69, 711–720.

Good, I.J. (1976). “On the Application of Symmetric Dirichlet distributions and their mixtures to contingency tables,” Ann. Statist., 4, 1159–1189.

Good, I.J. (1983a). Good Thinking: The Foundations of Probability and its Applications. Univ. of Minnesota Press.

Good, I.J. (1983b). “Who is a Bayesian?” Letter in American Statistician 37 (Feb.), 95.

Good, I.J. and Y. Mittal (1985). “The Amalgamation and Geometry of Two-by-Two Contingency Tables,” Ann. Statist., 15, 694–711.

Tukey, J.W., F. Mosteller, and S.E. Fienberg (1965). “Scoring Probability Forecasts." Department of Statistics, Harvard University, Memorandum NS-37.

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