Derivation of Bayes Theorem

1 Derivation

Following inspiration from Kruschke (2011).

2 From The Definition Of Conditional Probability:

\(P(A|B) = P(A,B) / P(B)\)

\(P(B|A) = P(A,B) / P(A)\)

3 Multiply Each Fraction By The Denominator:

\(P(A|B)P(B) = P(A,B)\)

\(P(B|A)P(A) = P(A,B)\)

4 Set The Two Expressions To Be Equivalent:

\(P(A|B)P(B) = P(B|A)P(A)\)

5 Divide by \(P(B)\):

\(P(A|B) = \frac{P(B|A)P(A)}{P(B)}\)