Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 772 — #780
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Chapter 18
Conditional Probability
given E,
Odds.H j E/ WWD Pr H j E
Pr H ˇˇ E
D Pr E j H PrŒH = PrŒE
Pr E ˇˇ
H PrŒH = PrŒE
D Pr E j H
Pr E ˇˇ PrŒH
H PrŒH
D Bayes-factor.E; H / Odds.H /;
(Bayes Theorem)
where
Bayes-factor.E; H / WWD Pr E j H
Pr E ˇˇ H
:
So to update the odds of H given the evidence E, we just multiply by Bayes Factor:
Lemma 18.9.2.
Odds for the TB test
Odds.H j E/ D Bayes-factor.E; H / Odds.H /:
The probabilities of test outcomes given in (18.6) and (18.7) are exactly what we
need to find Bayes factor for the TB test:
Bayes-factor.TB; pos/ D Pr pos j TB
Pr pos ˇˇ TB
D
D
1
1 Pr pos ˇˇ
TB
1
1 0:99 D 100:
So testing positive for TB increases the odds you have TB by a factor of 100, which
means a positive test is significant evidence supporting a diagnosis of TB. That
seems good to know. But Lemma 18.9.2 also makes it clear that when a random
person tests positive, we still can’t determine the odds they have TB unless we
know what are the odds of their having TB in the first place, so let’s examine that.
In 2011, the United States Center for Disease Control got reports of 11,000 cases
of TB in US. We can estimate that there were actually about 30,000 cases of TB