Does anyone know of a generalization of Bayes' theorem to multiple conditions?
From this answer I can see the definition of conditional probability with multiple conditions, but I couldn't find any reference to a statement of Bayes' theorem with several conditions. I only need a generalization to 2 conditions for what I'm working on, but I'm also curious to its statement in $n$ conditions.
Perhaps if we had the multiplication rule, too, we could derive Bayes' rule from the above?
One generally accepted definition for $\mathbb{P}\left(B\right) \neq 0$ is $$\mathbb{P}\left(A \mid B\right) = \dfrac{\mathbb{P}\left(A \cap B\right)}{\mathbb{P}\left(B\right)}\text{.}$$ If we had $n$ conditions, we could suppose that $\bigcup\limits_{i=1}^{\infty}A_i = A$, where $A_1, A_2, \dots$ is a partition of an event $A$, and $$\mathbb{P}\left(A_i \mid B_1 \cap B_2 \cap \cdots\cap B_n\right) = \dfrac{\mathbb{P}\left(A_i \cap B_1 \cap B_2 \cdots \cap B_n\right)}{\mathbb{P}\left(B_1 \cap B_2 \cdots \cap B_n\right)} = \dfrac{\mathbb{P}\left(B_1 \cap B_2 \cap \cdots\cap B_n \mid A_i\right)\mathbb{P}\left(A_i\right)}{\sum\limits_{A_i \subseteq A}\mathbb{P}\left(B_1 \cap B_2 \cap \cdots\cap B_n \mid A_i\right)\mathbb{P}\left(A_i\right)}\text{.}$$ [For someone who knows this material better than I do, feel free to let me know if I'm wrong somewhere. I'm basing this on intuition.] The formula above is particularly useful for introductory Bayesian methods (finding posterior distributions, in particular).
