to 24 times more likely to get lung cancer and, surprisingly, suffer
heart and circulatory diseases. Because heart disease was more
common than lung cancer (and indeed the No. 1 killer in the two
countries), Cornfield devised the famous 10-year Framingham
study. “Framingham gave him data about two groups of people,
those who had died of heart disease and those who had not,”
McGrayne says. “Within each group he had information about
seven risk factors. Applying Bayes’ theorem, he obtained a posterior
probability in the form of a logistic regression function, which
he then used to identify the four most important risk factors for
cardiovascular disease.”[ 15]
Cornfield also used Bayes’ theorem in clinical trials because
he discovered that Bayesian methods would enable him to reject
some hypotheses after only a small number of adverse observations. Frequentists would require the experiment to run to its
conclusion so they could gather the statistics needed for their tests
of significance. Amazingly, given Cornfield’s prestige (in 1974 he
became president of the American Statistical Association) and
the fact that he worked at National Institutes of Health (NIH), it
would be another 30 years before NIH regularly applied Bayes’
theorem to clinical trials.
Today actuaries use Bayesian analysis all the time. How could
we answer my CFO’s question concerning the Ebola risk without
making some prior assumptions? How could health actuaries price
for the individual health insurance exchanges in 2014 without
using all available data to make subjective assumptions? And, of
course, credibility is widely used in both pricing and reserving work
(see, for example, the January 2014 issue of the Astin Bulletin[ 16]
for a paper titled “Bayesian chain ladder models” for determining
incurred but not reported [IBNR] liabilities).
In a 1965 paper, professor Donald Jones, one of the authors
of Actuarial Mathematics, summed up the situation between
frequentists and Bayesians with the following metaphor. Recall
that frequentists need to run tests until they obtain a 95 percent
confidence interval. For each statistical test, the frequentist has a
different “box.” Jones writes,
“Each of his boxes has an intake hole at the top for the ex-
perimental data and an output hole at the right end. There
is a warning on each box which says: ‘Use seriously only
with k units or more of input.’ He would then offer you any
of his boxes. ... The Bayesian has only one black box—but
it has three holes: an additional intake hold on the left
for experience, information, and prior opinion. I think
actuaries should be among the most ardent welcomers and
users of the Bayesians’ one black box. In all their work and
training actuaries emphasize that decisions are based on
the data and judgment, but classical theories of statistics
have never offered a place for, or required, a well-defined
judgment factor.”[ 17]
It has been over 50 years since professor Jones made this ob-
servation. Bayesian analysis is much more accepted today because
( 1) it has been shown to work, repeatedly, and ( 2) there is a firm
mathematical basis for the theory and applications. Computer
software programs today apply Bayesian theory to enable computers
to “learn” and “make decisions,” such as translating a Wikipedia
article from English to Croatian. It is thought that the brain uses
Bayesian techniques to filter the hundreds of megabits of sensory
information that it receives every second.[ 18]
Over the last 150 years actuaries have been inventing or
adapting the latest techniques to make better judgments based
on the data at hand. We should be proud of the role actuaries
have played in promoting the use of Bayesian thinking. Let us
continue to stay on the cutting edge in order to better serve all
ROY GOLDMAN, MAAA, FSA, CERA, is a retired actuary
living in Jacksonville Beach, Fla.
[ 1] In 1982 Connecticut General Life Insurance Co CG purchased
INA, the first U.S. stock insurer, to form Cigna.
[ 2] E.J. Moorhead; Our Yesterdays: the History of the Actuarial
Profession in North America 1809-1979; p. 384.
[ 3] The full title: Observations on Reversionary Payments: On
Schemes for Providing Annuities for Widows, and for Persons in
Old Age: On the Method of Calculating the Values of Assurances on
Lives: and on the National Debt.
[ 4] S.B. McGrayne; The Theory That Would Not Die; p. 30.
[ 5] Ibid.; p. 33.
[ 6] Ibid.; p. 43.
[ 7] This quote is attributed by the author to Charles Hewitt, a
well-known casualty actuary. She also quotes Hewitt as referring
to Bailey as “the da Vinci or a Michelangelo” to the few actuaries
who understood his work.
[ 8] McGrayne op. cit.; p. 92.
[ 9] Ibid.; p. 100.
[ 10] Ibid.; p. 121.
[ 11] Howard Raiffa was told in the late 1940s that because he was
Jewish, he would be discriminated against as an engineer or
scientist. Because he had experienced discrimination in the army
and in housing, he believed what he was told. According to
McGrayne, “Then he learned that insurance actuaries were
graded on objective, competitive examinations. Seeking a field
where competence counted more than religion, Raiffa enrolled
in the University of Michigan’s actuarial program, where Arthur
Bailey had studied.”
[ 12] McGrayne writes that Daniel Wagner was so absent-minded that
his car once ran out of gas three times in a single day.
[ 13] On June 20, 2016, the New York Times reported that the United
States and Spain are still arguing over how to best clean up the
site. At the time, Spaniards and naval crewmen were mistakenly
told that there was no danger from radiation.
[ 14] In 1933 Cornfield began his career in the Department of Labor
because that was the one department that would hire Jews. The
author’s vignettes of Cornfield and all the other major
researchers make the book fun to read. They are all idiosyncratic;
some, like Fisher, are very cruel and crude people, but all are
[ 15] McGrayne op. cit.; p. 115.
[ 16] The Journal of the International Actuarial Association.
[ 17] Transactions of the Society of Actuaries; vol. 17; p. 33; as quoted by
Moorhead, op. cit.
[ 18] McGrayne op. cit.; p. 247–249.