I know how Bayes Theorem would look for 3 variables, but I'd like to know how to do this for n > 3 variables? For example, P(A|B,C,D,...N).
I'll include 3 variables here for reference if it helps with the explanation:
$$\def\P{\operatorname{\sf P}}\begin{align}\P(A\mid B,C) &= {\P(A,B,C)\over \P(B,C)} \\[1ex]&= {[\P(B\mid A,C)\cdot \P(A,C)]\over \P(B,C)} \\[1ex]&= {[\P(B\mid A,C)\cdot \P(A\mid C)\cdot \P(A)]\over \P(B,C)} \\[1ex]&= {[\P(B\mid A,C)\cdot \P(A\mid C)\cdot \P(A)]\over [\P(B|C)\cdot \P(C)]} \\[1ex]&= {[\P(B\mid A,C)\cdot \P(A\mid C)]\over \P(B\mid C)}\end{align}$$
So for I usually see Bayes as, if it has more than two variables I would use general Bayes Theorem:
Let $\Omega$ be the sample space and let $A_1, A_2, \cdots, A_n$ be non-empty disjoint events in our sample space such that $$A_1 \cup A_2\cup \cdots\cup A_j = \Omega$$
Using Bayes, this means $$P(A_j|B) = \frac{P(A_j \cap B)}{P(B)} = \frac{P(B|A_j)P(A_j)}{\sum_{i=1}^kP(B|A_i)P(A_i)} \qquad j=1, \cdots, k$$
Note: I more visual way to use Bayes is using probability trees. They are nice and straightforward, but can get overwhelming for more than three variables IMO.