Recent advances present the actuarial
profession with an opportunity to
strengthen its long-standing embrace of
Bayesian statistical methods.
imAGine tHAt you Are A pHysiciAn and a 45-year-old patient of yours just had
a positive mammogram. You also know the following facts:
■ ■ There is a 0.8 percent probability that a 40-to-50-year-old woman has breast cancer;
■ ■ If a woman has breast cancer, there is a 90 percent probability that she will have a positive
mammogram;
■ ■ If a woman does not have breast cancer, there is a 7 percent probability that she will have a
false positive mammogram.
What do you tell your patient is the probability that she actually has breast cancer?
The eminent cognitive scientist Gerd Gigeren- zer posed precisely this question to 48 physicians in Germany. Only four of them even came close to giving the right answer ( 9. 4 percent). Eight of them said that the probability was 90 percent: a tenfold overestimate. (See Page 40 for a solu- tion to this problem.) This example provides a window into a centuries-old debate in statistics that goes to the heart of how we reason from expe- rience and use data to make decisions. Its solution is an elementary exercise in Bayesian statistics, which, throughout much of the 20th century, was eclipsed by methods from the rival frequentist school. It is a credit to the actuarial pro- fession that Bayesian concepts—most notably credibility theory—have enjoyed a central place in ac- tuarial work for nearly a hundred years. Recent advances in Bayesian methods, however, have outpaced the actuarial profession’s embrace of them. Thanks to both computational and conceptual advances, Bayesian methods have undergone a dramatic renaissance in the past two decades. As a result, the 21st century is likely to see Bayesian methods gain promi- nence in a number of disciplines. These developments, should actuaries choose to embrace them more widely, will have a salutary effect on the profession. Bayesian methods are at once powerful and philosophically co- herent. And, as this example illustrates, they sometimes have vital significance.
a Bayesian Update
Bayesian statistics dates from 1763, when the mathematician Richard Price presented a paper written by his
late friend, the Rev. Thomas Bayes, to the British Royal
Society. It is likely that Bayes wrote his paper in reaction to the problem of induction as posed a few decades
earlier by the philosopher David Hume. Namely, why do
we expect the future to resemble the past?
At the heart of Bayesian inference is the notion that
probabilities represent degrees of belief about uncertain statements. Examples of such statements include:
■ The patient I am speaking with has cancer;
■ Barack Obama will be re-elected in 2012;
■ There will be a magnitude 6. 7 or greater earthquake
in the San Francisco Bay Area before 2030;
■ The ultimate losses for a cohort of insurance claims
incurred in 2010 will be in the $1 million-to-$1.2 million range;
■ The coin I am tossing has a limiting relative frequency
(“true probability”) of heads between 0.498 and 0.502.
The methodological centerpiece of Bayesian inference is the idea that probabilities about uncertain
statements are continually updated in light of empirical data. In the first example, the physician’s degree of
belief in the proposition that the patient has cancer increased from 0.008 to .094 because of the positive test
result. Were a second, independent piece of evidence
(such as a history of smoking) to emerge, this probability would shift again from 0.094 to something higher
still. No single piece of evidence is decisive, and neither
should it be used in a vacuum. Rather, each piece plays
