Thinking and Deciding

BAYES’S THEOREM 123
Formulas for Bayes’s theorem
Let us now see how this sort of problem can be described by a general formula,
derived algebraically from the rules of coherence and other assumptions. 6 In the
medical example, if we use H to indicate the hypothesis (possibility) that the patient
has cancer and D to indicate the datum (fact, evidence) of a positive test result,
we would have the values p(H) =.10; p(D|H) =.90; p(D| ∼H) =.20. The
two conditional probabilities say that the probability of the datum, given that the
hypothesis is true, is .90 and the probability of the datum, given that the hypothesis
is false (∼ H), is .20. (The expression ∼ A means “not A” or“A is false,” for any
proposition A. p(∼A) is therefore “the probability that A is false.”)
What we want to know in this case is (H|D). That is, we know that D is true,
and we want to use this evidence to revise our probability judgment for the disease.
Because we know that D is true, we can conditionalize on D. In other words, we
can look at the probability of H in just those cases where D is known to be true —
that is, the 27 patients.
We can calculate (H|D) from the multiplication rule, p(H & D) =p(H|D) ·
p(D), which implies:
p(H & D)
p(H|D) = (5.1)
p(D)
In our example, p(H & D), the probability of the disease and the positive test result
together, is .09 (9 out of 100 cases), and p(D), the probability of the test result is
.27 (9 cases from cases who have cancer and 18 from cases who do not, out of 100
cases. This is what we concluded before: The probability of cancer is 1/3, or .33.
Why is this formula correct? Recall that the conditional probability of H given
D is the probability that we would assign to H if we knew D to be true. If we think
6 We shall not discuss the mathematical details here. They are found in many textbooks of probability.