From Algorithms to Z-Scores - matloff - University of California, Davis

2.6. BAYES’ RULE 13
Note by the way that events involving X2 are NOT independent of those involving X1. For instance,
we found in (2.16) that
yet from (2.14) we have
2.6 Bayes’ Rule
P (X1 = 1|X2 = 2) = 0.43 (2.17)
P (X1 = 1) = 0.48. (2.18)
(This section should not be confused with Section 12.5. The latter is highly controversial, while
the material in this section is not controversial at all.)
Following (2.16) above, we noted that the ingredients had already been computed, in (2.13) and
(2.12). If we go back to the derivations in those two equations and substitute in (2.16), we have
P (X1 = 1|X2 = 2) = P (X1 = 1 and X2 = 2)
P (X2 = 2
(2.19)
=
P (X1 = 1 and X2 = 2)
P (X1 = 1 and X2 = 2) + P (X1 = 2 and X2 = 2)
(2.20)
=
P (X1 = 1)P (X2 = 2|X1 = 1)
P (X1 = 1)P (X2 = 2|X1 = 1) + P (X1 = 2)P (X2 = 2|X1 = 2) (2.21)
Looking at this in more generality, for events A and B we would find that
P (A|B) =
P (A)P (B|A)
P (A)P (B|A) + P (not A)P (B|not A)
(2.22)
This is known as Bayes’ Theorem or Bayes’ Rule. It can be extended easily to cases with
several terms in the denominator, arising from situations that need to be broken down into several
subevents rather than just A and not-A.
2.7 ALOHA in the Notebook Context
Think of doing the ALOHA “experiment” many, many times.