p(theory)
data
q(theory)
The Theory That Would Not Die
by Sharon Bertsch McGrayne (Yale University Press, New Haven, 2011)
DURiNG A ROUTiNe MiDAiR ReFUeLiNG in January 1966, a U.S. B- 52 bomber armed with four
hydrogen bombs brushed against the nozzle of a fuel tanker. Forty thousand gallons of fuel burst into flames,
killing seven of the crew members and raining tons of aircraft parts—and four hydrogen bombs—on the
isolated Spanish town of Palomares. None of the bombs exploded, and three of the four bombs were located
within 24 hours. But the fourth bomb remained elusive. Strategic Air Command needed to use all available
information to locate the wayward nuclear bomb as soon as possible.
Thus begins one of the many gripping, and occasionally startling, stories
that grace Sharon Bertsch McGrayne’s
highly enjoyable new history of Bayesian
inference, The Theory That Would Not Die.
McGrayne’s account is capacious enough
to include vignettes as disparate as the
search for lost nuclear weapons, the freeing of Alfred Dreyfus from Devil’s Island,
the successful effort to crack the Nazis’
Enigma cipher machine, determining the
authorship of the Federalist Papers, and
the founding of the Casualty Actuarial Society (CAS).
bayesian Logic
Bayes’ “inverse probability” rule concerns
the updating of probabilities about uncertain statements in light of empirical data.
In modern terms, it takes the form of a simple conditional probability calculation:
p(theory)
data
q(theory) = p(theory | data) =
p(data | theory)p(theory)
. p(data)
Here p(theory) and q(theory) represent one’s “prior” and
“posterior” probabilities of the truth of an uncertain statement before and after the receipt of a body of relevant data.
The second equality in the above expression is an elementary
fact of probability theory known as Bayes’ theorem. But the
real substance of Bayesian inference is encapsulated in the
first equality: q(theory) = p(theory | data). This is a philosophical assumption, not a mathematical fact. And it encapsulates
the central Bayesian principle that conditional probability is
the mechanism by which new data should be combined with
background beliefs or knowledge to make predictions and
inferences. (See our article, “Enhanced Credibility,” in the Sep-tember-October 2010 issue of Contingencies for more details.)
This is the logic that the Navy’s top scientist, John Craven,
developed to guide the search for the lost nuclear bomb near
Palomares and later used to find the nu-
clear submarine USS Scorpion, lost at sea
in May 1968. The area of search is divided
into a grid, each cell of which is assigned a
probability of containing the lost object. The
search is prioritized in the areas of highest
probability, and Bayes’ theorem is used to
update these probabilities in the light of fail-
ing to find the object in the areas that have
been searched. This procedure incorpo-
rates the probability of finding the object in
a location X if it actually is in that location.
Bayesian updating reflects that fact that
failing to find the lost object in, say, a deep
water area, conveys different information
than failing to find it in an open field. Mc-
Grayne (in a private conversation with us)
and others have speculated reasonably that
Bayesian search probably was used in the
detection of Osama bin Laden in Abbottabad, Pakistan.
theoretical origins
A striking aspect of the history of Bayesian inference is the large
number of times it independently has been rediscovered. The
rule is of course named after the Rev. Thomas Bayes, who first
outlined the logic of inverse probability in a paper that was
posthumously presented to the British Royal Society in 1763 by
his friend Richard Price.
Bayes’ work relates interestingly to both philosophy and
actuarial science. McGrayne astutely links Bayes’ work with
the philosophical discussions of cause and effect that were
“in the air” subsequent to the publication of David Hume’s
landmark Treatise on Human Nature. And indeed Price used
Bayes’ rule to dispute Hume’s skeptical thesis (today echoed
in Nassim Taleb’s Black Swan) that no amount of experience
is sufficient to establish a cause-and-effect relationship. Price,
interestingly, also was a founding father of the actuarial profession—he consulted for the Equitable Life Assurance Society of
