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.