# Differences

This shows you the differences between two versions of the page.

bayes_theorem [2013/03/30 20:00] jongbor [Decompose a conditional probability into a product] |
bayes_theorem [2014/04/19 15:41] (current) |
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$$ P(A,B) = P(A|B)P(B) = P(B|A)P(A) $$ | $$ P(A,B) = P(A|B)P(B) = P(B|A)P(A) $$ | ||

- | This can be rearranged as: | + | This can be rearranged as the **product rule**: |

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

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

- | Luckily, this is the same as $P(A,B) = P(A)P(B|A)$, but adding $C$ in the conditional part of the probability. | + | Luckily, this is the same as the product rule $P(A,B) = P(A)P(B|A)$, but adding $C$ in the conditional part of the probability. |

==== Generalization to $n$ variables ==== | ==== Generalization to $n$ variables ==== |