# Bayes Rule

A Bayes Rule is a probability update rule which states that you must multiply the prior probability (that a belief is true) by the probability that the evidence is true given that the belief is true divided by the probability that the evidence is true regardless of whether the belief is true.

• AKA: Bayes Theorem.
• Context:
• It can be stated as,if $E_1,E_2,\dots,E_n$ are mutually disjoint events with a priori probabilities $P(E_i)\neq 0,(i=1,2,\dots,n)$ then for any arbitrary event $A$ which is a subset of $\displaystyle\bigcup_{i=1}^{n}E_i$, such that $P(A)\gt 0$, we have the posterior probabilities $P(E_i|A)=\frac{P(E_i)P(A|E_i)}{\displaystyle\sum_{i=1}^{n}P(E_i)P(A|E_i)},i=1,2,\dots,n.$ Here $P(A|E_i),i=1,2,\dots,n$ are called likelihoods.
• It can be used by a Bayesian Inference Algorithm.
• It can be proved by application of the Product Rule.
• It can be used as a Decision Rule based on minimizing Average Loss.
• It can be restated as “The plausibility of your belief depends on the degree to which your belief -- and only your belief--explains the evidence for it. The more alternative explanations there are for the evidence, the less plausible your belief is.
• Example(s):
• Posterior Probability = The Prior × Normalized Likelihood.
• $Pr(A \vert B) = \frac{Pr(B \vert A)}{1} \times \frac{Pr(A)}{Pr(B)}$.
• In answering a question on a multiple choice test, a student either knows the answer (with probability $p$) or he guesses (with probability $1-p$).Assume that the probability of answering a question correctly is unity for a student who knows the answer and $\frac{1}{m}$ for the student who guesses, where $m$ is the number of multiple choice alternatives.Supposing a student answers a question correctly, the probability that he really knows the answer can be found out by using the Bayes theorem as follows, let $E_1=$The student knows the answer, $E_2=$The student guesses the answer and $A=$The student answers correctly.Then $P(E_1)=p,P(E_2)=1-p,P(A|E_1)=1$ and $P(A|E_2)=\frac{1}{m}$.Now using Bayes theorem, the probability that a student really knows the answer given that the student answers it correctly is $P(E_1|A)=\frac{P(E_1)P(A|E_1)}{P(E_1)P(A|E_1)+P(E_2)P(A|E_2)}=\frac{p.1}{p.1+(1-p).\frac{1}{m}}=\frac{mp}{1+(m-1)p}$
• Counter-Example(s):
• See: Belief Revision; Probability Theory; Bayesian Network; Naive-Bayes Model; Naive-Bayes Classifier; Bayesian Probability; Bayesianist; Bayesian Methods; Bayesian Network; Bayesian Model Selection, Bayes Networks, Statistical Proof.