Conditional Independence
- $X$ is conditionally independent of $Y$ given
$Z$ if the probability distribution governing $X$ is
independent of the value of $Y$ given the value of $Z$; that
is, if
\[ \forall_{x_i,y_j,z_k} P(X = x_i \,|\, Y = y_j, Z = z_k) = P(X = x_i \,|\, Z
= z_k) \] more compactly, we write \[ P(X \,|\, Y,Z) = P(X \,|\, Z) \]
- For example, $Thunder$ is conditionally independent of
$Rain$, given $Lightning$
\[ P(Thunder \,|\, Rain, Lightning) = P(Thunder \,|\, Lightning) \]
- Naive Bayes uses cond. indep. to justify
\[ \array{
P(X,Y\,|\,Z) &= & P(X\,|\,Y,Z)\cdot P(Y\,|\,Z) \\
&= & P(X\,|\,Z)\cdot P(Y\,|\,Z)
} \]
José M. Vidal
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