Product rule of probabilities and conditioning

Question

I have a probably very basic question regarding the product rule for probabilities. If we have a two probability densities say $p(a|b)$ and $p(b|c,d)$, is $p(a|b)p(b|c,d) = p(a,b | c,d)$? What has me slightly confused is that the second term is conditioned on variables $c$ and $d$ and so I am not sure the product rule applies directly here. I would have thought that the first term $p(a|b)$ would need be conditional on $c$ and $d$ explicitly also.

Would someone mind clarifying this for me please?

Answer 1

$$P(a|b) P(b|c,d ) = P(a,b | c,d) \\ \iff \frac{P(a\cap b)}{P(b)} \times P(b\cap c\cap d) = P(a\cap (b\cap c\cap d))$$

Suppose $a$ and $b$ are independent and $P(b\cap c\cap d)\ne 0$. Then this means $$P(a) P(b\cap c\cap d) = P(b\cap c\cap d) \cdot P(a | b\cap c\cap d) \\ \iff P(a)=P(a|b\cap c\cap d)$$ This is true if and only if $a$ and $b\cap c\cap d$ are also independent, but this is not necessarily true. Therefore, the identity you propose isn’t necessarily true.

Answer 2

In terms of events you are actually asking:

Do we have: $$P(A\cap B)\times P(B\cap C\cap D)=P(B)\times P(A\cap B\cap C\cap D)\text{ ?...}$$

Certainly not in general.