Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 16

1.6.2 Bayes' Rule

Let the space S be subdivided into N mutually exclusive events B j , j ¼ 1, , N .

The probabilities of events B j are called a priori probabilities because they repre-
sent the probabilities of B j before the experiment is performed. Similarly, the
probabilities P { A | B j } are typically known prior to conducting the experiment, and
are called transition probabilities .
Now, suppose that the experiment is performed and, as a result, event A
occurred. The probability of the occurrence of any of the events B j —knowing
that the event A has occurred-is called a posteriori probability PfB j jAg .Thea
posteriori probability is calculated using conditional probability ( 1.43 ) and the
theorem of total probability ( 1.65 ) resulting in:

...

P f AB j g
PfAg ¼

PfAjB j gPfB j g
P

PfB j jAg¼ : (1.70a)

N
i¼ 1 PfAjB i gPfB i g

The formula in ( 1.70a ) is known as Bayes' rule .
Because ( 1.70a ) is a complicated formula, some authors like Haddad [HAD06,
p. 32] consider it is better to write Bayes' rule using the following two equations:

P f AB j g
PfAg ¼

PfAjB j gPfB j g
PfAg

PfB j jAg¼

PfAg¼ X

N

PfAjB i gPfB i g: (1.70b)

i¼ 1

1.7 Independent Events

Consider two events A and B with the nonzero probabilities P { A } and P { B }, and
that the occurrence of one event does not affect the occurrence of the other event.
That means that the conditional probability of event A , given B , is equal to the
probability of event A ,

PfAjBg¼PfAg: (1.71)

Similarly, the conditional probability of event B , given A , is equal to the
probability of event B ,

PfBjAg¼PfBg: (1.72)