Bayes’ theorem is central to probability. It serves as the foundation for statistical inference, which is used to estimate population parameters based on observations. For example, Bayes’ theorem may be applied to find the probability that a coin is fair, given that all prior flips have landed on heads.
Mathematically, Bayes’ theorem may be stated as follows for events \(A\) and \(B\):
\[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\]
In the above, \(P(A)\) is called the prior, and represents the probability of \(A\) before evidence is incorporated. Likewise, \(P(A|B)\) is called the posterior, and represents the probability of \(A\) after evidence is incorporated.
It is easy to arrive at this theorem from the following two formulations of the definition of conditional probability:
\[P(A|B) = \frac{P(A \cap B)}{P(B)}\] \[P(A \cap B) = P(A)P(B|A)\] \[\Downarrow\] \[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\]