Bayes Theorem

Published June 23, 2026

The same joint event can be factored as

\mathbb{P}(B)\mathbb{P}(A\mid B)

or as

\mathbb{P}(A)\mathbb{P}(B\mid A)

Bayes theorem is a rule for reversing conditional probabilities. It tells us how to compute the probability of a hypothesis after observing evidence.

Let

A

be a hypothesis and let

B

be an observed event. The conditional probability of

A

given

B

\begin{equation*}\mathbb{P}(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}, \qquad \mathbb{P}(B)>0.\end{equation*}

The terms in the formula have standard names:

\begin{equation*}\underbrace{\mathbb{P}(A\mid B)}_{\text{posterior}} =\frac{ \underbrace{\mathbb{P}(B\mid A)}_{\text{likelihood}} \underbrace{\mathbb{P}(A)}_{\text{prior}}}{\underbrace{\mathbb{P}(B)}_{\text{evidence}}}.\end{equation*}

The prior is the probability of

A

before seeing

B

BAYES THEOREM

Let

A

and

B

be events with

\mathbb{P}(A)>0

and

\mathbb{P}(B)>0

. Then

\begin{equation*}\mathbb{P}(A\mid B) = \frac{\mathbb{P}(B\mid A)\mathbb{P}(A)}{\mathbb{P}(B)}.\end{equation*}

The proof is only the definition of conditional probability used twice. Since

\begin{equation*}\mathbb{P}(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}\quad \text{and} \quad\mathbb{P}(B\mid A)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(A)},\end{equation*}

we get

\begin{equation*}\mathbb{P}(A\cap B)=\mathbb{P}(B\mid A)\mathbb{P}(A).\end{equation*}

Substitution into the first identity gives

\begin{equation*}\mathbb{P}(A\mid B)=\frac{\mathbb{P}(B\mid A)\mathbb{P}(A)}{\mathbb{P}(B)}.\qquad \Box\end{equation*}

References