Say we have a situation where A depends on B, and B depends on C. If I'm not mistaken, this means that A is independent of C given B (even though B depends on C, A given B sort of indirectly incorporates that already- that's how I understand it)
And we have the probability functions for
P(A|B) and P(B|C) and P(A,B|C)
There should be a way to get P(A|C) directly. My first guess is that P(A,B|C) can be changed into P(A|C), but I'm not sure how.
It makes sense intuitively since we have a chain of dependence, but I don't know where to start with the math. I tried using the chain rule and rearranging
$$P(A,B|C) = \frac{P(C|A,B)P(A|B)P(B)}{P(C)}$$
But I'm not sure how to rearrange that further
I think that you can't establish a relationship between event A and event C from the given information. The dependency and independency between event A and event C might occur. You can utilize Venn diagram to illustrate both cases. The given information indicates $\text{A} \!\cap\!\text{B} \neq \varnothing$ and $\text{B} \!\cap\!\text{C} \neq \varnothing$. Now your goal is to check $\text{A} \!\cap\!\text{C} \stackrel{?}{=} \varnothing$ from the given information. Using Venn diagram, the dependency and independency between event A and event C might occur. For the later case, see the below picture
Now is it possible to show the former case? Yes, it is. See the below picture,
Notice that for both cases, $\text{A} \!\cap\!\text{B} \neq \varnothing$ and $\text{B} \!\cap\!\text{C} \neq \varnothing$ hold.