Against the Gods: The Remarkable Story of Risk (26 page)

De Moivre's gift to mathematics was an instrument that made it
possible to evaluate the probability that a given number of observations
will fall within some specified bound around a true ratio. That gift has
provided many practical applications.

For example, all manufacturers worry that defective products may
slip through the assembly line and into the hands of customers. One
hundred percent perfection is a practical impossibility in most instancesthe world as we know it seems to have an incurable habit of denying us
perfection.

Suppose the manager of a pin factory is trying to hold down the
number of defective pins to no more than 10 out of every 100,000 produced, or 0.01% or the total.14 To see how things are going, he takes a random sample of 100,000 pins as they come off the assembly line and
finds 12 pins without heads-two more than the average of 10 defectives that he had hoped to achieve. How important is that difference?
What is the probability of finding 12 defective pins out of a sample of
100,000 if, on the average, the factory would be turning out 10 defective
pins out of every 100,000 produced? De Moivre's normal distribution
and standard deviation provide the answer.

But that is not the sort of question that people usually want
answered. More often, they do not know for certain before the fact
how many defective units the factory is going to produce on the average.
Despite good intentions, the true ratio of defectives could end up
higher than 10 per 100,000 on the average. What does that sample of
100,000 pins reveal about the likelihood that the average ratio of defectives will exceed 0.01% of the total? How much more could we learn
from a sample of 200,000? What is the probability that the average ratio of defectives will fall between 0.009% and 0.011%? Between .007%
and .013%? What is the probability that any single pin I happen to pick
up will be defective?

In this scenario, the data are given-10 pins, 12 pins, 1 pin-and the
probability is the unknown. Questions put in this manner form the
subject matter of what is known as inverse probability: with 12 defective
pins out of 100,000, what is the probability that the true average ratio
of defectives to the total is 0.01%?

One of the most effective treatments of such questions was proposed by a minister named Thomas Bayes, who was born in 1701 and
lived in Kent." Bayes was a Nonconformist; he rejected most of the
ceremonial rituals that the Church of England had retained from the
Catholic Church after their separation in the time of Henry VIII.

Not much is known about Bayes, even though he was a Fellow of
the Royal Society. One otherwise dry and impersonal textbook in statistics went so far as to characterize him as "enigmatic."16 He published
nothing in mathematics while he was alive and left only two works that
were published after his death but received little attention when they
appeared.

Yet one of those papers, Essay Towards Solving A Problem In The
Doctrine Of Chances, was a strikingly original piece of work that immortalized Bayes among statisticians, economists, and other social scientists.
This paper laid the foundation for the modern method of statistical
inference, the great issue first posed by Jacob Bernoulli.

When Bayes died in 1761, his will, dated a year earlier, bequeathed
the draft of this essay, plus one hundred pounds sterling, to "Richard
Price, now I suppose a preacher at Newington Green."17 It is odd that
Bayes was so vague about Richard Price's location, because Price was
more than just a preacher in Islington in north London.

Richard Price was a man with high moral standards and a passionate belief in human freedom in general and freedom of religion in
particular. He was convinced that freedom was of divine origin and
therefore was essential for moral behavior; he declared that it was better
to be free and sin than to be someone's slave. In the 1780s, he wrote a
book on the American Revolution with the almost endless title of
Observations on the Importance of the American Revolution and the Means of
Making it a Benefit to the World in which he expressed his belief that the
Revolution was ordained by God. At some personal risk, he cared for
the American prisoners of war who had been transferred to camps in
England. Benjamin Franklin was a good friend, and Adam Smith was an
acquaintance. Price and Franklin read and criticized some of the draft
chapters of The Wealth of Nations as Smith was writing it.

One freedom bothered Price: the freedom to borrow. He was
deeply concerned about the burgeoning national debt, swollen by the
wars against France and by the war against the colonies in North
America. He complained that the debt was "funding for eternity" and
dubbed it the "Grand National Evil."18

But Price was not just a minister and a passionate defender of human
freedom. He was also a mathematician whose work in the field of probability was impressive enough to win him membership in the Royal
Society.

In 1765, three men from an insurance company named the Equitable
Society called on Price for assistance in devising mortality tables on which
to base their premiums for life insurance and annuities. After studying the
work of Halley and de Moivre, among others, Price published two articles on the subject in Philosophical Transactions; his biographer, Carl Cone, reports that Price's hair is alleged to have turned gray in one night of
intense concentration on the second of these articles.

Price started by studying records kept in London, but the life
expectancies in those records turned out to be well below actual mortality rates.19 He then turned to the shire of Northampton, where
records were more carefully kept than in London. He published the
results of his study in 1771 in a book titled Observations on Reversionary Payments, which was regarded as the bible on the subject until well
into the nineteenth century. This work has earned him the title of the
founding father of actuarial science-the complex mathematical work
in probability that is performed today in all insurance companies as the
basis for calculating premiums.

And yet Price's book contained serious, costly errors, in part because
of an inadequate data base that omitted the large number of unregistered
births. Moreover, he overestimated death rates at younger ages and
underestimated them at later ages, and his estimates of migration into
and out of Northampton were flawed. Most serious, he appears to have
underestimated life expectancies, with the result that the life-insurance
premiums were much higher than they needed to be. The Equitable
Society flourished on this error; the British government, using the same
tables to determine annuity payments to its pensioners, lost heavily.20

Two years later, after Bayes had died, Price sent a copy of Bayes's
"very ingenious" paper to a certain John Canton, another member of the
Royal Society, with a cover letter that tells us a good deal about Bayes's
intentions in writing the paper. In 1764, the Royal Society subsequently
published Bayes's essay in Philosophical Transactions, but even then his
innovative work languished in obscurity for another twenty years.

Here is how Bayes put the problem he was trying to solve:

PROBLEM

Given that the number of times in which an unknown event has
happened and failed: Required the chance that the probability of its
happening in a single trial lies somewhere between any two degrees
of probability that can be named.21

The problem as set forth here is precisely the inverse of the problem as
defined by Jacob Bernoulli some sixty years earlier (page 118). Bayes is
asking how we can determine the probability that an event will occur
under circumstances where we know nothing about it except that it has
occurred a certain number of times and has failed to occur a certain
number of other times. In other words, a pin could be either defective
or it could be perfect. If we identify ten defective pins out of a sample of
a hundred, what is the probability that the total output of pins-not just
any sample of a hundred-will contain between 9% and 11% defectives?

Price's cover letter to Canton reflects how far the analysis of probability had advanced into the real world of decision-making over just a
hundred years. "Every judicious person," Price writes, "will be sensible that the problem now mentioned is by no means a curious speculation in the doctrine of chances, but necessary to be solved in order to
[provide] a sure foundation for all our reasonings concerning past facts,
and what is likely to be hereafter."22 He goes on to say that neither
Jacob Bernoulli nor de Moivre had posed the question in precisely this
fashion, though de Moivre had described the difficulty of reaching his
own solution as "the hardest that can be proposed on the subject of
chance."

Bayes used an odd format to prove his point, especially for a dissenting minister: a billiard table. A ball is rolled across the table, free to stop
anywhere and thereafter to remain at rest. Then a second ball is rolled
repeatedly in the same fashion, and a count is taken of the number of
times it stops to the right of the first ball. That number is "the number of
times in which an unknown event has happened." Failure-the number
of times the event does not happen-occurs when the ball lands to the
left. The probability of the location of the first ball-a single trial-is to
be deduced from the "successes" and "failures" of the second.23

The primary application of the Bayesian system is in the use of new
information to revise probabilities based on old information, or, in the
language of the statisticians, to compare posterior probability with the
priors. In the case of the billiard balls, the first ball represents the priors
and the continuous revision of estimates as to its location as the second
ball is repeatedly thrown represents the posterior probabilities.

This procedure of revising inferences about old information as new
information arrives springs from a philosophical viewpoint that makes Bayes's contribution strikingly modem: in a dynamic world, there is no
single answer under conditions of uncertainty. The mathematician
A.F.M. Smith has summed it up well: "Any approach to scientific
inference which seeks to legitimise an answer in response to complex
uncertainty is, for me, a totalitarian parody of a would-be rational
learning process."24

Although the Bayesian system of inference is too complex to recite
here in detail, an example of a typical application of Bayesian analysis
appears in the appendix to this chapter.

The most exciting feature of all the achievements mentioned in this
chapter is the daring idea that uncertainty can be measured. Uncertainty
means unknown probabilities; to reverse Hacking's description of certainty, we can say that something is uncertain when our information is
correct and an event fails to happen, or when our information is incorrect and an event does happen.

Jacob Bernoulli, Abraham de Moivre, and Thomas Bayes showed
how to infer previously unknown probabilities from the empirical
facts of reality. These accomplishments are impressive for the sheer
mental agility demanded, and audacious for their bold attack on the
unknown. When de Moivre invoked ORIGINAL DESIGN, he
made no secret of his wonderment at his own accomplishments. He
liked to turn such phrases; at another point, he writes, "If we blind
not ourselves with metaphysical dust we shall be led by a short and
obvious way, to the acknowledgment of the great MAKER and
GOUVERNOUR of all."25

We are by now well into the eighteenth century, when the Enlightenment identified the search for knowledge as the highest form of human
activity. It was a time for scientists to wipe the metaphysical dust from
their eyes. There were no longer any inhibitions against exploring the
unknown and creating the new. The great advances in the efforts to
tame risk in the years before 1800 were to take on added momentum
as the new century approached, and the Victorian era would provide
further impulse.