Bayes Theorem with multiple variables.

Question

I have seen some various forms of Bayes Theorem, I was wondering if we regard $X = (x_1, x_2, x_3,...,x_n)$ and $Y = (y_1, y_2, y_3, ..., y_m)$ where $X$ and $X$ consist of continuous distributions.

Can we say that:

$f(X|Y) = \frac{f(Y|X)*f(X)}{f(Y)}$

More generally, how does one define conditional distribution when you condition on multiple variables at the same time?

Answer 1

For non singular marginal densities

$f(y_1,y_2,...,y_n | x_1,x_2,...,x_n)=\frac{f(y_1,y_2,...,y_n ,x_1,x_2,...,x_n)}{f(x_1,x_2,...,x_n)}$

So in your case $f(X)=f(x_1,x_2,...,x_n)$ is the marginal density of $X$ and similarly

$f(x_1,x_2,...,x_n | y_1,y_2,...,y_n)=\frac{f(y_1,y_2,...,y_n ,x_1,x_2,...,x_n)}{f(y_1,y_2,...,y_n)}$

So in your case $f(Y)=f(y_1,y_2,...,y_n)$ is the marginal density of $Y$