A conditional probability is expressed as for any two
events
and is called the ``probability of , given
.'' Its definition is

(9.4) 
Two events, and , are called independent if and only if
; otherwise, they are called dependent. An important and sometimes misleading concept is
conditional independence. Consider some third event,
. It might be the case that and are dependent, but when
is given, they become independent. Thus,
; however,
. Such examples
occur frequently in practice. For example, might indicate
someone's height, and is their reading level. These will
generally be dependent events because children are generally shorter
and have a lower reading level. If we are given the person's age as
an event , then height is no longer important. It seems intuitive
that there should be no correlation between height and reading level
once the age is given.
The definition of conditional probability, (9.4),
imposes the constraint that

(9.5) 
which nicely relates to . This results in
Bayes' rule, which is a convenient way to swap and :

(9.6) 
The probability distribution, , is referred to as the prior, and is the posterior.
These terms indicate that the probabilities come before and after
is considered, respectively.
If all probabilities are conditioned on some event,
,
then conditional Bayes' rule arises, which only differs from
(9.6) by placing the condition on all probabilities:

(9.7) 
Steven M LaValle
20200814