Definition 2.4 (Conditional Independence): \(X\) is independent of \(Z\) conditioning on \(Y\), if

\[ \begin{aligned} p(x, y, z) &= \\ & \begin{cases} {p(x,y)p(y,z) \over p(y)} = p(x,y)p(z|\,y) & \text{if }p(y) > 0 \\ 0 & \text{otherwise.} \\ \end{cases} \end{aligned} \]

Note:

\[ \begin{aligned} {p(x,y)p(y,z) \over p(y)} = p(x,y)p(z|\,y) = p(x|\,y)p(y,z) = p(x)p(y|\,x)p(z|\,y) \end{aligned} \]

Source: Information Theory.

# posted by rot13(Unafba Pune) @ 5:57 AM