Conditional Independence

The Random Variable Z1Z_1 is conditionally independent of Z2Z_2 given ZZ if

P(Z1,Z2∣Z)=P(Z1∣Z)⋅P(Z2∣Z)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(Z1∣Z2,Z)=P(Z1∣Z)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