Conditional Independence

Eq. $\eqref{eq1}$ means that “the probability of $A$ given $C$ is the same as the probability of $A$ given both $B$ and $C$, this equality expresses that $B$ contributes nothing to the certainty of $A$”¹ and hence $A$ and $B$ are conditionally independent, given $C$. Eq. $\eqref{eq2}$ better expresses this meaning, and we can derive Eq. $\eqref{eq2}$ from Eq. $\eqref{eq1}$¹:

Besides, it should be noted that, the fact that $A$ and $B$ are independent doesn’t means that they are conditionally independent given event $C$, like an example in video²:

From the right case we can see that $P(A,B)=P(A)P(B)$, but $P(A,B\vert C) = P(A\vert C)P(B\vert C)$, i.e. Eq. $\eqref{eq2}$, doesn’t hold.

By the way, the left Venn diagram can explain Eq. $\eqref{eq2}$ very well, but not clearly for Eq. $\eqref{eq1}$.

References

• Review Notes and Supplementary Notes CS229 Course Machine Learning Standford University, chapter 2: Probability Theory Review for Machine Learning, p. 5.