# Is joint conditionally independent equivalent to marginally conditionally independent？

## Heading ##I am wondering whether these two properties are equivalent:

• $$X$$ is conditionally independent of $$Y$$ given $$Z$$

• $$X$$ is conditionally independent of $$Y$$ given $$a^T Z$$, $$\forall a \in R^p$$

Thank you.

• Just my guess: This is equivalently asking the sigma algebra generated by them is identical or not. Since both $a$ and $Z$ are $p$-dimensional, you may pick a group of $\{a_1, a_2, \ldots, a_p\}$ which are linearly independent, then the set $\{a_1^TZ = z_1, a_2^TZ = z_2, \ldots a_3^TZ = z_3\}$ will uniquely gives the solution of the form $Z = z$, so they should be equivalent. – BGM Nov 21 '18 at 9:55
• I also have thought of this, but I cannot proof the result strictly.Thank you. – Aaron Nov 22 '18 at 15:36

Suppose $$X = Z_1+Z_2+\epsilon_1,Y = Z_1+Z_2+\epsilon_2, Z=(Z_1,Z_2)^T,\epsilon_1 \bot Z,\epsilon_2 \bot Z$$. It's obvious that $$X \bot Y |Z$$. But if we take $$a=(1,0)^T$$ , $$X \not \bot Y|a^TZ$$. So maybe the first property is a lot weaker than the second property.