Today, we learn how to hedge bets. The University
of Houston's College of Engineering presents this
series about the machines that make our
civilization run, and the people whose ingenuity
created them.
Your wife and her friend went
out and got you a white dog for your birthday, and
you wonder which of them selected it. At first
blush, it'd be a fifty-fifty guess. But you know
two things: your wife doesn't like white dogs very
much, and her friend likes
them a lot. So, the friend probably chose the
dog.
We can actually do a calculation here, but it's not
simple. If the likelihood of your wife's picking a
white dog is fifteen percent, and her friend's
doing so is ninety percent, the odds that her
friend chose it turn out to be eighty-five percent.
To get that answer, we use something called
Bayesian statistics -- named after
eighteenth-century nonconformist cleric Thomas
Bayes. Bayes' first book, written in 1731, was on
Divine Benevolence. Five years later he
wrote a second, quite different, book. In it, he
defended Newton's calculus against an attack by the
British philosopher, Berkeley.
Not long afterward, Bayes was made a member of the
Royal Society. He didn't write much more. But, two
years after he died, the Royal Society published
his paper on The Doctrine of Chances. In
it, he suggests using prior knowledge to improve
our prediction of outcomes. That's why, given your
wife's and her friend's preferences, you're pretty
sure who picked that white dog.
Another more serious example: Suppose you're tested
for a certain cancer. The test has a five percent
error rate. But prior knowledge tells you that only
one person in a hundred thousand really has the
cancer. That means a positive reading is almost
certainly wrong. The test is nearly useless.
But Bayes' ideas have had tough sledding --
probably because they can be so easily misused. You
can misinterpret your friend's preferences. Maybe I
mislead myself when I say it's completely unlikely
that I have that cancer. You can't really argue
with the math, but you have to be very careful
with the premises.
All this came to my attention when the New York
Times did an article with the title,
Subconsciously, Athletes May Play Like
Statisticians. A good athlete, it seems, uses
a vast wealth of subjective statistics. A tennis
player calls up her knowledge of an opponent with a
particularly dangerous backhand, and she adjusts
the percentages of her own shots to compensate.
The Times article describes a Bayesian
thought process as what we use when "uncertainty
becomes great enough to give past experience an
edge over current observation." Of course that can
be very dicey. Think about the roulette player who
knows perfectly well that the house odds are
stacked against him and still says, "Oh yes, but
red likes me."
By now, the obvious usefulness of Bayesian
statistics has triumphed over the equally obvious
dangers that go with its use. For we know we can
get into serious trouble by dropping our guard when
we temper our statistics with what we only
think is true.
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.
(Theme music)
For a nice textbook treatment of Bayesian statistics,
see: D. P. Bertsekas and John N. Tsitsiklis,
Introduction to Probability. Belmont, MA:
Athena Scientific, 2002, Section 1.4. The white dog
calculation (above) was made using Bayes' Rule on
page 31. (Section 1.3, by the way, includes a nice
discussion of Monty Hall Problem -- see Episode 1577.)
For more on Bayesian statistics see: http://www.abelard.org/briefings/bayes.htm
I. hacking, Bayes, Thomas. Dictionary of
Scientific Biography (C.C. Gilespie, ed.), New
York: Charles Scribner's Sons, 1970-1980, Vol. I,
pp. 531-532.
D. Leonhardt, Subconsciously, Athletes May Play
Like Statisticians. The New York Times, Science
Times, Tuesday, January 20, 2004, pg. 1 and 6.
For some more technical discussions of Bayesian statistics,
(and dogs) see: http://webuser.bus.umich.edu/plenk/index.htm
I am most grateful to three UH colleagues for very
helpful counsel on this episode: Charles Peters,
Mathematics, as well as Jagannatha Rao & David
Zimmerman, Mechanical Engineering. Just for the fun
of it, here's another example developed by Dr.
Peters:
You've received a Christmas present with no tag on
it. It had to've come from either your girlfriend
or your brother. Since you know your brother is
Christmas-challenged, you'd give 2 to 1 odds that
he was the one who forgot to label the package.
(This is your prior information.)
You also know that 50% of all presents given by
your brother are ugly ties. Your girlfriend has
better taste and only 5% of her presents are ugly
ties. (This is your model for the outcome, given
the state of nature.) Thus it would seem that, if
the present is an ugly tie, the liklihood that it
came from your brother is 0.50/0.05 or ten to
one.
You open the present and, sure enough, it
is an ugly tie. So now you wonder just
what the odds are that the present came from your
brother? One formulation of Bayes' rule says that
the posterior odds equal the product of the prior
odds times the likelihood ratio. In this case,
the prior odds were 2/1 and the likelihood ratio
is 10/1. Thus, when we add our posterior
knowledge, the odds are = 2 X 10 = 20/1. In other
words, we've increased the likelihood of that
ugly tie being your brother's gift, from 10/1, to
20/1.
The Engines of Our Ingenuity is
Copyright © 1988-2003 by John H.
Lienhard.
