The rule of conditional probability states: $$ P(A|B) = \frac{P(A,B)}{P(B)} $$

However, it is not clear to me how you can/must condition the random variables with more than just two. The top answer on this post states that you are a free to condition your variables as you like: Marginalization of conditional probability with the conditional probability rule generalized to multiple variables:

$$ P(x_1,...,x_n|y_1,...,y_m)=\frac{P(x_1,...,x_n,y_1,...,y_m)}{P(y_1,...,y_m)} $$

Is that true? I could not find this generalized rule anywhere to be honest.

So when applying this to the product rule, am I allowed to do:

$$ P(A,B,C,D) = P(A,B|C,D)*P(C,D) $$


$$ P(A,B,C,D) = P(A,B,C|D)*P(D) $$


$$ P(A,B,C,D) = P(A|B,C,D)*P(B,C,D) $$


1 Answer 1


Yes, that is true. You can for example think of $\mathcal X=(B,C,D)$ as a random vector, write $$P(A,\mathcal X)=P(A|\mathcal X)P(\mathcal X)=P(\mathcal X|A)P(A)$$


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