Conditional independence does not always imply independence: for example, independent random variables are rarely conditionally independent on their total or maximum.
Conditional independence does not imply independence; conditionally independent random variables uniform on (0,u) where u is uniform on (0,1), for example, are not independent.
Furthermore, If (X,Y,Z) are (mutually) independent, then X is conditionally independent of Y. Proof:
fx,y|z=fx,y,z/fz=fxfyfz/fz=fxfy=fx|zfy|z.
Elevate your skills with our comprehensive Machine Learning Course.