Let X,Y,Z be discrete RVs taking value in a state space S. X and Y are conditionally independent given Z if P(X=x,Y=y∣Z=z)=P(X=x∣Z=z)P(Y=y∣Z=z) for any x,y,z such that P(Z=z)>0.
The following two statements are equivalent: 1. X and Y are conditionally independent given Z. 2. For any x,y,z so that P(Y=y,Z=z)>0 P(X=x∣Y=y,Z=z)=P(X=x∣Z=z) 3. For any x,y,y′,z so that P(Y=y,Z=z)>0 and P(Y=y′,Z=z)>0 P(X=x∣Y=y,Z=z)=P(X=x∣Y=y′,Z=z)