So, I've got a few questions:
$1)$ is $P(a,b) = P(b,a)$ ?
$2)$ How do I get some intuition for Bayes rule? I know don't really understand what is happening. $P(h|d) = (P(d | h) P(h) / P(d))$. I get the top half of the equation, we're finding the probability that d is occurring given h, but what does the bottom "do"? I think of multiplication as two events occurring in sequence, but what does a divide do?
$3)$ I don't really understand what a Bayes rule with background knowledge ($b$) is derived from. It seems to be a pretty esoteric equation that my professor gave me:
$$\mathbb P(h|d, b)=\frac{\mathbb P(d|h,b)\mathbb P(h|b)}{\mathbb P(d|b)}.$$ None of this is for homework.