Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

2.11 BAYES' FORMULA

Bayes' formula is a useful equation that is used to compute conditional probabilities, or derive marginal probabilities by conditioning on events.

Theorem 2.2 (Bayes' formula). For events with :

(2.69)

Likewise, for :

(2.70)

Proof. From the definition of conditional probability, we can write the following two expressions:

(2.71)

Combining these results and solving for P(A|B) or P(B|A) yields Bayes' formula, which is also called Bayes' rule.

Example 2.43 (Binary symmetric channel). An important example of Bayes' formula is the binary symmetric channel (BSC) shown in Figure 2.10, which is a model for bit errors that occur in a digital communication system. For a binary system, the transmitted symbol has two outcomes: we will use {0, 1} to represent the two binary elementary events. This model is equivalent to the coin-toss experiment with and . As shown in Chapter 3, ...