(^528) 18. PROBABILITY AND STATISTICS
Statistics has been the focus of metaphysical debate, just like mathematics. For
some early thinkers, such as Augustus de Morgan, the applications of probability
were simply a matter of human ignorance: If we knew any reason for a system to
be in one state rather than another, we would posit that reason as a physical law.
In the absence of such a reason, all possible states are equally likely. A principle
very close to this one is the basis of the second law of thermodynamics as now
deduced in statistical physics. Yet other thinkers took a different point of view,
positing some resemblance of the future to the past. This principle is the basis of
the "frequentist" philosophy, which asserts that the probability of a future event is
to be hypothesized from its occurrence in the past. The standard example is the
question "What is the probability that the sun will rise tomorrow?" Assuming that
we have adequate records that would have noted any exceptions to this very regular
event over the past 5,000 years, a purely frequentist statistician would offer odds
of 1,800,000 to 1 in favor of the event happening tomorrow. However, as William
Feller (1906-1970) pointed out in his classic textbook of probability, our records
really do not guarantee that there have been no exceptions. Our confidence that
the sun rose in the remote past is based on the same considerations that give us
confidence that it will rise tomorrow.
Opposed to the frequentists are the Bayesians, who believe it is possible to
assign a probability to an event before a similar event has occurred. Classical
probabilists, with their urn models, drawings from a deck of cards, and throws
of dice, were in effect Bayesians who believed that symmetry considerations and
intuition enabled people to assign probabilities to hypothetical events. The results
of experiment, where available, helped to revise those assignments through Bayes'
theorem: // the events A\,...,An are mutually exclusive and exhaustive, and  is
any event, then
P(A LM - P^fcAi?) - P{Ak)P{B\Ak)
nAk\B)- p[B) -Ó.Ñ(âì.)Ñ(Ë.)·
The use of this formula is as follows. From some basic principles of symmetry,
or purely subjectively, the events Ak are assigned hypothetical probabilities. Then
the conditional probability of event  is computed assuming each of these events.
After an experiment in which event  occurs, the probability of Ak is "updated"
to the value of Ñ(Áê\Â) computed from this formula. The simplest illustration is
the case of a chest containing two drawers. Drawer 1 contains two gold coins, and
drawer 2 contains a silver coin and a gold coin. The two events are Á÷: drawer 1
is chosen; A 2 : drawer 2 is chosen. Each is assigned a preliminary probability of .
Event B is "a gold coin is drawn from the drawer." The conditional probabilities
are easily seen to be P{B\A{) = 1, Ñ{Â\Á·é) = . If in fact a gold coin is drawn,
then
I. é É ï
Ñ{Áé\Â)and_2 _ 2_ _ f
1 2 + 2 2 4 J1 1
= T-T-^TT = i = \
1 " Ï ô 22 ô
2.4. Correlations and statistical inference. The many intricate techniques
that statisticians have developed today for analyzing data to determine if two vari-
ables are correlated are far too complex to be discussed in full here. But we cannot