```prob ← cond ##.bayes prior                  ⍝ Bayes' formula

Bayesian Statistics using a Fork by Steve Mansour
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Suppose the probability that a person has cancer is 3%.   A certain test will be
positive 90% of the time when a person has cancer.   But there is a 2% chance of
a false positive. What is the probability that a person actually has the disease
if the result is positive?  We know  P(Test|Disease) = 0.9.   What we are trying
to find is P(Disease|Test).

We use the conditional rule: P(A|B) = P(A∩B)/P(B) and P(B|A) = P(A∩B)/P(A).

From this we can show:

P(A|B) P(B) = P(A∩B) = P(B|A) P(A)

We can then show that: P(B|A) = P(A|B)P(B)/P(A).

The marginal probability P(A) = SUMi[P(A∩Bi)] = SUMi[P(A|Bi)P(Bi)]

This allows us to derive Bayes' Formula:

P(A|Bi)P(Bi)
P(Bi|A) = -------------------
SUMi[P(A|Bi)P(Bi)]

Let us first set the prior probabilities.

We can create a vector: P(Cancer), P(No Cancer)

C←0.03                  ⍝ Probability of Cancer

⎕←PRIOR←C,1-C           ⍝ Prior probabilities
0.03 0.97

Now let us set the conditional probabilities.
Again, we create a vector: P(Positive|Cancer), P(Postive|No Cancer)

COND←0.9 0.02           ⍝ Conditional probabilities

Now let find the Bayesian probabilities.
The result will be a vector: P(Cancer|Positive),P(No Cancer|Positive).

Observe that Bayes' Formula above consists of a product divided by an inner pro-
duct. Let us define the function bayes as a fork:

bayes ← × ÷ +.×         ⍝ times div sumProduct

We the use the conditional probabilities as the left argument and the prior
probabilities as the right argument:

COND bayes PRIOR        ⍝ Bayes Formula
0.5819 0.4181

The answer is quite surprising. Given that the test is positive, the probability
of cancer is less than 60%  which means there is greater than a 40%  chance that
the patient does not have cancer.

Bayes' formula  can  be applied to situations where there are more than two out-
comes.   Consider  the  following table showing the breakdown of students by Sex
(F=Female,M=Male) and Party (D=Democrat, I=Independent, R=Republican):

frequency show D.Sex D.Party

Count     |         D         I         R |   Total
------------------------------------------|--------
F         |         3         2         4 |       9
M         |         8         9        12 |      29
------------------------------------------|--------
Total     |        11        11        16 |      38

Suppose we know that Pr(Sex|Party)  what is Pr(Party|Sex)? From Bayes' Formula
we need the prior probabilities for Female:

⎕←PRIORF←11 11 16÷38        ⍝ Pr(Party)
0.28947 0.28947 0.42105

We also need the conditional probabilities for female given party:

⎕←CONDF←3 2 4÷11 11 16      ⍝ Pr(F|D),Pr(F|I),Pr(F|R)
0.27273 0.18182 0.25

Applying Bayes' formula, we obtain a vector of posterior probabilities:

CONDF bayes PRIORF          ⍝ Pr(D|F),Pr(I|F),Pr(R|F)
0.33333 0.22222 0.44444

Observe that we can obtain the same result by dividing the joint frequencies by
the marginal frequency for female:

⎕←3 2 4÷9                   ⍝ Pr(Party|Female)
0.33333 0.22222 0.44444

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