genomic, drug, and social science studies that reject hypotheses
of “there is no effect” at the 95 percent level. We expect that
one in 20 of the conclusions of these published studies results
from nothing more than pure chance. Indeed, the phenomenon
might be even more widespread than this if scientists have a
tendency to test multiple hypotheses and publish only the ones
resulting in low p-values. It is important to keep this fact in
mind when reading about apparent medical breakthroughs or
newly discovered disease risk factors in the news.
does probability exist?
The philosophical premise that probability represents degrees
of belief rather than frequencies of random events therefore has
enormous practical consequences. But many statisticians and
scientists resist Bayesian statistics, feeling it injects an inappropriate degree of subjectivity into what should be an “objective”
scientific process. Given the somewhat polemical tone of some
Bayesian writings, this feeling is perhaps understandable. The
Italian probabilist Bruno de Finetti, arguably the most important
Bayesian theorist of the 20th century, famously opened his book on
probability theory with the capitalized pronouncement, “
PROBABILITY DOES NOT EXIST.” It is notable that de Finetti began his
career working as an actuary at an insurance company in Trieste.
De Finetti’s pronouncement is a colorful way of saying that
probabilities represent states of belief rather than facts about the
world. But for many scientists, this severely limits the usefulness
of Bayesian statistics. For example, the U.S. Geological Survey
once estimated that there is a 70 percent chance of a 6. 7 or higher
magnitude earthquake in the San Francisco Bay Area before the
year 2030. Recall David Freedman’s objection: Interpreting the
USGS’s statement as a de Finettian statement of opinion amounts
to “changing the subject.” After all, we are interested in California’s faults, not the personal beliefs of a group of geologists.
But this objection rests on a false distinction. To be sure,
these so-called subjective Bayesian probabilities do represent
the geologists’ beliefs. But these beliefs reflect the best available
theory, background knowledge, scientific judgment, and data.
Yes, they are subjective, but only in the sense that is true of all
scientific models. The late Princeton philosopher Richard Jef-
frey summed it up this way:
Your “subjective” probability is not something fetched
out of the sky on a whim; it is what your actual judgment
should be, in view of your information to date and other
people’s information.
For this reason, Jeffrey observed that “judgmental probability” would be a less misleading term than “subjective
probability.” Jeffrey’s comment is helpful because judgment
must be used in any scientific field that involves model selection. No less a scientist than Albert Einstein recognized this
form of subjectivity in science when he famously remarked,
“Physical concepts are free creations of the human mind.”
Judgment must be used when selecting frequentist likelihood
functions as well as Bayesian prior probability functions. Selecting prior probabilities is a form of judgmental model selection,
a process that is not unique to Bayesianism.
let’s Be More frequent Bayesians
The implication of these arguments is not that classical methods
should be avoided at all cost. Non-Bayesian approaches are helpful
in at least two sorts of common situations. For example, an actuary analyzing the claims experience in a large population of young
male and female drivers, or policyholders with poor, average, and
good credit, would arrive at essentially the same risk relativities
regardless of the prior probability with which he or she started. In
such data-rich situations, frequentist statistics is sufficient.
Second, actuaries increasingly face situations, such as fraud
detection or customer segmentation, in which approaches with
less structure are appropriate. In such cases, creative data visualization and inventive approaches to learning from large data
sets, often powered by modern machine-learning algorithms,
can be more fruitful than starting with likelihood functions and
priors. John Chambers, the eminent statistician who invented
the S programming language, called this approach “greater
statistics,” which he defined as “everything related to learning
from data.” We believe that this is very much in the spirit of
statistician John Tukey’s philosophy of exploratory data analysis (EDA). In short, a pluralistic attitude is wise, and actuaries
should evaluate claims about the superiority of Bayesian statistics from a pragmatic frame of reference.
With that said, modern Bayesian methods currently are gaining prominence in a wide swath of the sciences with impressive
speed, and actuaries are in an excellent position to capitalize
on these developments. Actuarial science has a deep Bayesian
heritage relative to the other professions. Bayes’ friend Price
did pioneering work in annuity pricing for the Society for Equitable Assurances on Lives and Survivorships. De Finetti began
his career as an actuary. The Bayesian notion of “shrinking”
a purely likelihood-based estimate toward a complementary
source of information was implicit in actuarial credibility theory for much of the 20th century because of the pioneering work
of such actuaries as A. W. Whitney, Arthur Bailey, and LeRoy Simon. Hans Bühlmann, another actuary turned academic, made
the connection explicit with his theory of Bayesian credibility.
The best reason for actuaries to embrace the Bayesian method, ultimately, is to become more effective professionals and
better serve the public interest. This can occur along a number
of avenues. For example:
■ ■ Rigorous foundations—Mirroring the larger divide in
statistics, there are competing frequentist and Bayesian versions of credibility theory. Our view is that the frequentist
“limited-fluctuation” theory of credibility obscures the inherently Bayesian nature of credibility methods. As with
credibility theory, the goal of Bayesian statistics is to combine background information or expert knowledge with the
data at hand. On the other hand, the core idea of the limited
