Conditional Independence

The Random Variable $Z_1$ is conditionally independent of $Z_2$ given $Z$ if $P\left(Z_1,Z_2\mid Z\right)=P\left(Z_1\mid Z\right)\cdot P\left(Z_2\mid Z\right)$

and we get $P\left(Z_1\mid Z_2,Z\right)=P\left(Z_1\mid Z\right)$ So we can drop parts of the right side when we have conditional independence. This is heavily exploited in Bayesian Networks.

• Bayes Rule with multiple evidence
• dimensionaloty reducion
• drop probabilities that we alread know via causal things in the world
• clever ordering can make computations more efficient
• seen quite often, Independence not so much