Proof of Bayes Theorem
In two minutes
Bayes theorem is one of the most important formulas in statistics and is used in a variety of settings like clinical trials, speech detection, and self-driving cars.
Why is Bayes theorem useful?
Simply put, Bayes theorem(and, as an extension, Bayes factor) helps us update our existing beliefs based on newly acquired data
Bayes theorem does this by helping us represent one conditional probability in terms of another conditional probability that we already know or can calculate easily.
Conditional Probability
To understand the Bayes theorem, we’ll first need to understand conditional probabilities. A conditional probability is the measure of the probability of an event B occurring, given that an event A has already occurred.
Derivation of Bayes theorem
Bayes theorem uses the fact that, in probability, P(A and B) can also be written as P(B and A). If you remember from my last post, for independent events, the following is always true:
Bayes theorem is the clever application of the above rule on dependent events using their respective conditional probabilities.
Bayes theorem states:
where:
Bayes theorem can also be expanded to use multiple dependent events. Say, if A is dependent on three mutually exclusive events B, C, and D, then, P(A) is given as:
