Determining Conditional Independence from Marginal Independence?

Question

so if I have a 3 columns of binary variables X, Y and Z with their respective values and I would like to determine whether X,Y are conditionally independent given Z. How can I go about doing this? Also how can I incorporate marginal independence (of two variables) into this equation?

Note: format is as such

X Y Z
1 0 1
0 0 1
1 1 0
etc...

and I want to output true if x,y are conditionally independent given z and false otherwise and I can test for marginal independence if I want to...

I know that P(Y|Z) - P(Y|X,Z) = 0 is one way but how would I calculate those values from the above...

Thank you.

Answer 1

"Conditional on Z" means within levels of Z. So for each level of Z, figure out whether X and Y are independent. X and Y are independent if the proportion of Y values that are equal to 1 is the same in each level of X.

Symbolically, $X \perp Y|Z \implies P(Y=1|X=1, Z=z) = P(Y=1|X=0, Z=z), \forall Z$

Some R/pseudocode:

c.ind = TRUE
for (z in unique(Z)) {
  if (mean(Y[Z==z & X==0]) != mean(Y[Z==z & X==1)) c.ind = FALSE
}
return(c.ind)