Given this table, is X conditionally independent of Y give Z?
I know conditional independence is P(X|Y,Z) = P(X, Z), but I'm a bit confused about how I can use that with values from the table. I understand how to get P(X|Y,Z) from the table, but am a bit confused about the right hand side of the equation, for example P(X = 0, Z = 0) can have 2 values? I think I'm missing something.
It's a self-study question, so let me give you some hints so that you could solve it by yourself.
The table shows the joint probabilities $\sum_{x,y,z} P(x, y, x) = 1$.
Recall the definition of conditional probability
$$ P(A|B) = \frac{P(A \cap B)}{P(B)} $$
it will be needed to compute the conditional probabilities from the joint probabilities.
You also need the law of total probability
$$ P(A) = \sum_n P(A \cap B_n) $$
to get the marginal probabilities.
Use the above and check against the definition of conditional independence
$$ P(A|B,C) = P(A|C) $$
[...] In this case, $A$ and $B$ are said to be conditionally independent given $C$
Thought I'd post an answer of my question since I think I have the correct answer now in case others see this and think it's useful.
Following Tim's advice I calculated P(X|Z) and compared that to P(X|Y,Z) which I think is equivalent to P(X^Z)P(Y^Z)
so using the table in the original question:
P(X=0|Z=0) = P(X=0^Z=0)/P(Z=0) = (1/15 + 1/10) / (1/3) = 1/2
similarly:
P(X=1|Z=0) = P(X=1^Z=0)/P(Z=0) = 1/2
P(X=0|Z=1) = P(X=0^Z=1)/P(Z=1) = 2/3
P(X=1|Z=1) = P(X=1^Z=1)/P(Z=1) = 1/3
For X to be conditionally independent of Y given Z then P(Y^Z) should be 1, but we can see that it isn't. Therefore X isn't conditionally independent of Y give Z.
This also seems to just be clear when looking at the table, since we need to know the value of Y and Z to get X.
