Forms

Events

Simple form

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

P(A|B) \propto P(A) \cdot P(B|A).

P(A|B) = c \cdot P(A) \cdot P(B|A) \ and \ P(\neg A|B) = c \cdot P(\neg A) \cdot P(B|\neg A)\cdot

c = \frac{1}{P(A) \cdot P(B|A) + P(\neg A) \cdot P(B|\neg A) }.

Extended form

P(B) = {\sum_j P(B|A_j) P(A_j)}, \implies P(A_i|B) = \frac{P(B|A_i)\,P(A_i)}{\sum\limits_j P(B|A_j)\,P(A_j)}\cdot

P(A|B) = \frac{P(B|A)\,P(A)}{ P(B|A) P(A) + P(B|\neg A) P(\neg A)}\cdot

Random variables

Simple form

f_X(x|Y=y) = \frac{P(Y=y|X=x)\,f_X(x)}{P(Y=y)}.

P(X=x|Y=y) = \frac{f_Y(y|X=x)\,P(X=x)}{f_Y(y)}.

f_X(x|Y=y) = \frac{f_Y(y|X=x)\,f_X(x)}{f_Y(y)}.

Extended form

f_Y(y) = \int_{-\infty}^\infty f_Y(y|X=\xi )\,f_X(\xi)\,d\xi .

Bayes’ rule

O(A_1:A_2|B) = O(A_1:A_2) \cdot \Lambda(A_1:A_2|B)

\Lambda(A_1:A_2|B) = \frac{P(B|A_1)}{P(B|A_2)}

O(A_1:A_2) = \frac{P(A_1)}{P(A_2)}

O(A_1:A_2|B) = \frac{P(A_1|B)}{P(A_2|B)}