# Finding if $X$ and $Y$ are independent, given a list of variables $Z = [Z_1,\ldots,Z_n]$

I am running the pc-algorithm for causal networks. In this algorithm, I need to verify if two arrays $$X$$ and $$Y$$ are independent given a set of arrays $$Z_1,\ldots, Z_n$$. In papers, it is usually assumed gaussian independence tests, which I am trying not to use in favor of a more generic approach. I tried to use mutual information, but as can be seen in my discussion of yesterday, mutual information doesn't seem to be a good approach.
Question.Is there a rule or property which allows me to evaluate if $$X$$ and $$Y$$ are independent given $$Z = Z_1,\ldots, Z_n$$? All sources I found so far concern only the case in which $$Z$$ has the same dimension as $$X$$ and $$Y$$.

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• Sorry, as stated your problem is completely unintelligible. Otherwise, if as in mathoverflow.net/questions/364546/… $Z = X + Y$ with (there) independent random vectors $X$ and $Y$, then if $(X$ and $Y$ are not "too simple" (f.i. constant), then $X$ and $Y$ are not conditionally independent. I think that's not what you want to know, but what? The pc-algorithm is completely irrelevant in this situation. – Dieter Kadelka yesterday