What is Bayes’ Rule?

What is Bayes’ Rule?

The Bayes’ Rule helps us to enhance our estimations of the unconditional probability of another event by using knowledge about the outcome of one event. It is regarded as the cornerstone of Bayes’ inference approach to statistical inference. This approach uses the unconditional probabilities of A and B and information about the conditional probability of A given B to compute the probability of B given A.

Example of Bayes’ Rule:
$ P(A|B) = \frac{P(A|B) × P(A) P(B)}{P(B)} $
Where,
A , B = events
P(A|B)= probability of A given B is true
P(A), P(B)= the independent probabilities of A and B

Bayes’ rule has many financial and risk management applications. For example, suppose that 10% of fund managers are superstars. Superstars have a 20% chance of beating their benchmark by more than 5% each year, whereas regular fund managers have only a 5% chance of winning their benchmark by more than 5%. Suppose there is a fund manager that beats her benchmark by 7%. What is the possibility that she is a superstar?
Here, event A is the manager significantly outperforming the fund’s benchmark. Event B is that the manager is a superstar.
Using Bayes’ rule:

$ Pr(Star| High Return)= \frac{Pr(High Return | Star) Pr(Star)}{Pr(High Return)} $

The probability of a superstar is 10%. The probability of a high return if she is a superstar is 20%. The probability of a high return from either type of manager is Pr(High Return) = Pr(High Return]Star) Pr(Star) + Pr(High Return | Normal) Pr(Normal) = 20% x 10% + 5% x 90% = 6.5%

Why is Bayes’ Rule important?

Bayes’ rule explains how information from one occurrence can be used to learn about another event. It is based on basic probability concepts. Because analysts are frequently interested in revising their views based on observed facts, this is a useful, practical tool.