I am wondering how I would apply Bayes rule to expand an expression with multiple variables on either side of the conditioning bar.
In another forum post, for example, I read that you could expand $P(a,z \mid b)$ using Bayes rule like this (see Summing over conditional probabilities): $$P( a,z \mid b) = P(a \mid z,b) P(z \mid b)$$
However, directly using Bayes rule to expand $P(a,z \mid b)$ doesn't seem to be the right way to start out:
$$P(a,z\mid b) = { P(b\mid a,z)P(a,z) \over P(a)}$$