Editing Independence (probability theory) (section)

===For random variables===
Intuitively, two random variables <math>X</math> and <math>Y</math> are conditionally independent given <math>Z</math> if, once <math>Z</math> is known, the value of <math>Y</math> does not add any additional information about <math>X</math>. For instance, two measurements <math>X</math> and <math>Y</math> of the same underlying quantity <math>Z</math> are not independent, but they are conditionally independent given <math>Z</math> (unless the errors in the two measurements are somehow connected).

The formal definition of conditional independence is based on the idea of [[conditional distribution]]s. If <math>X</math>, <math>Y</math>, and <math>Z</math> are [[discrete random variable]]s, then we define <math>X</math> and <math>Y</math> to be conditionally independent given <math>Z</math> if

:<math>\mathrm{P}(X \le x, Y \le y\;|\;Z = z) = \mathrm{P}(X \le x\;|\;Z = z) \cdot \mathrm{P}(Y \le y\;|\;Z = z)</math>

for all <math>x</math>, <math>y</math> and <math>z</math> such that <math>\mathrm{P}(Z=z)>0</math>. On the other hand, if the random variables are [[Continuous random variable|continuous]] and have a joint [[probability density function]] <math>f_{XYZ}(x,y,z)</math>, then <math>X</math> and <math>Y</math> are conditionally independent given <math>Z</math> if

:<math>f_{XY|Z}(x, y | z) = f_{X|Z}(x | z) \cdot f_{Y|Z}(y | z)</math>

for all real numbers <math>x</math>, <math>y</math> and <math>z</math> such that <math>f_Z(z)>0</math>.

If discrete <math>X</math> and <math>Y</math> are conditionally independent given <math>Z</math>, then

:<math>\mathrm{P}(X = x | Y = y , Z = z) = \mathrm{P}(X = x | Z = z)</math>

for any <math>x</math>, <math>y</math> and <math>z</math> with <math>\mathrm{P}(Z=z)>0</math>. That is, the conditional distribution for <math>X</math> given <math>Y</math> and <math>Z</math> is the same as that given <math>Z</math> alone. A similar equation holds for the conditional probability density functions in the continuous case.

Independence can be seen as a special kind of conditional independence, since probability can be seen as a kind of conditional probability given no events.