Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 771 — #779
18.9. Probability versus Confidence 771
seems to suggest that it’s worth betting with high odds that you have TB, because
it makes sense to bet against something unlikely happening—like the test being
wrong. But having TB actually turns out to be a lot less likely than the test being
wrong. So the either-or of Corollary 18.9.1 is really an either-or between something
happening that is extremely unlikely—having TB—and something that is
only very unlikely—the diagnosis being wrong. You’re better off betting against
the extremely unlikely event, that is, it is better to bet the diagnosis is wrong.
So some knowledge of the probability of having TB is needed in order to figure
out how seriously to take a positive diagnosis, even when the diagnosis is given with
what seems like a high level of confidence. We can see exactly how the frequency
of TB in a population influences the importance of a positive diagnosis by actually
calculating the probability that someone who tests positive has TB. That is, we want
to calculate Pr TB j pos , which we do next.
18.9.2 Updating the Odds
Bayesian Updating
A standard way to convert the test probabilities into outcome probabilities is to use
Bayes Theorem (18.2). It will be helpful to rephrase Bayes Theorem in terms of
“odds” instead of probabilities.
If H is an event, we define the odds of H to be
Odds.H / WWD PrŒH
PrŒH D
PrŒH
1 PrŒH :
For example, if H is the event of rolling a four using a fair, six-sided die, then
PrŒroll four D 1=6; so
Odds.roll four/ D 1=6
5=6 D 1 5 :
A gambler would say the odds of rolling a four were “one to five,” or equivalently,
“five to one against” rolling a four.
Odds are just another way to talk about probabilities. For example, saying the
odds that a horse will win a race are “three to one” means that the horse will win
with probability 1=4. In general,
PrŒH D Odds.H /
1 C Odds.H / :
Now suppose an event E offers some evidence about H . We now want to find
the conditional probability of H given E. We can just as well find the odds of H