# Bayes' Theorem/General Result

## Theorem

Let $\Pr$ be a probability measure on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\set {B_1, B_2, \ldots}$ be a partition of the event space $\Sigma$.

Then, for any $B_i$ in the partition:

$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A} = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\sum_j \condprob A {B_j} \map \Pr {B_j} }$

where $\ds \sum_j$ denotes the sum over $j$.

## Proof

Follows directly from the Total Probability Theorem:

$\ds \map \Pr A = \sum_i \condprob A {B_i} \map \Pr {B_i}$

and Bayes' Theorem:

$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A}$

$\blacksquare$

## Source of Name

This entry was named for Thomas Bayes.