Bayes' Rule(Bayes' Theorem, Bayes' Law)

1. Overview

In probability theory and statistics, Bayes’ theorem (alternatively Bayes’ law or Bayes’ rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person's age can be used to more accurately assess the probability that they have cancer than can be done without knowledge of the person’s age.

One of the many applications of Bayes’ theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in Bayes’ theorem may have different probability interpretations. With the Bayesian probability, interpretation the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for availability of related evidence. Bayesian inference is fundamental to Bayesian statistics.

2. Description

2.1 Prerequisite

2.1.1 Conditional probability 

$$P(A\: |\: B)=\frac{P(A\cap B)}{P(B)}$$

2.1.2 Multiplication rule

$$P(A\cap B)=P(A\: |\: B)\times P(B)$$

2.2 Definition

$$P(A\: |\: B)=\frac{P(B\: |\: A)\times P(A)}{P(B)}$$

3. Example

3.1 Example 1

Even without a direct causal link, there exist some arguments to support such claims

3.2 Example 2

3.3 Example 3

4. Reference

https://en.wikipedia.org/wiki/Bayes%27_theorem

https://en.wikipedia.org/wiki/Bayesian_inference