Conditional Independence

$$P(A|B,C)=P(A|B)$$

Also

$$P_{X,Y,Z}(x,y,z)=P_{X|Z}(x|z)P_{Y|Z}(y|z)P_Z(z)$$

This means that when controlling for the effects of $B$ then $C$ doesn’t have any new information about $A$ (they are independent)

Conditional independence doesn’t mean that two events are independent. They are only independent given information about $B$. They still may be strictly dependent.

Notation

$$A\perp C|BorX\perp Y|Z$$

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References