Editing Conditional independence (section)

===Decomposition===
: <math>
X \perp\!\!\!\perp Y \mid Z
\quad \Rightarrow \quad
h(X) \perp\!\!\!\perp Y \mid Z
</math>

'''Proof'''
From the definition of conditional independence, we seek to show that:
: <math>
X \perp\!\!\!\perp Y \mid Z
\quad \Rightarrow \quad
P(h(X), Y \mid Z) = P(h(X) \mid Z) P(Y \mid Z)
</math>
. The left side of this equality is:
: <math>
P(h(X)=a, Y=y \mid Z=z) = \sum_{X \colon h(X)=a} P(X=x, Y=y \mid Z=z)
</math>
, where the expression on the right side of this equality is the summation over <math>X</math> such that <math>h(X)=a</math> of the conditional probability of <math>X, Y</math> on <math>Z</math>.
Further decomposing,
: <math>
\begin{align}
\sum_{X \colon h(X)=a} P(X=x, Y=y \mid Z=z) =& \sum_{X \colon h(X)=a} P(X=x \mid Z=z) P(Y=y \mid Z=z) \\
=& P(Y=y \mid Z=z) \sum_{X \colon h(X)=a} P(X=x \mid Z=z) \\
=& P(Y \mid Z) P (h(X) \mid Z)
\end{align}
</math>
. Special cases of this property include
* <math>
(X, W) \perp\!\!\!\perp Y \mid Z
\quad \Rightarrow \quad
X \perp\!\!\!\perp Y \mid Z
</math>
** '''Proof:''' Let us define <math> A = (X,W) </math> and <math> h(\cdot) </math> be an 'extraction' function <math> h(X,W) = X</math>. Then:
: <math>
\begin{align}
(X,W) \perp\!\!\!\perp Y \mid Z
\quad &\Leftrightarrow \quad
A \perp\!\!\!\perp Y \mid Z \\
&\Rightarrow \quad
h(A) \perp\!\!\!\perp Y \mid Z \quad &\text{Decomposition} \\
&\Leftrightarrow \quad
X \perp\!\!\!\perp Y \mid Z
\end{align}
</math>
* <math>
X \perp\!\!\!\perp (Y, W) \mid Z
\quad \Rightarrow \quad
X \perp\!\!\!\perp Y \mid Z
</math>
** '''Proof:''' Let us define <math> V = (Y,W) </math> and <math> h(\cdot) </math> be again an 'extraction' function <math> h(Y,W) = Y</math>. Then:
: <math>
\begin{align}
X \perp\!\!\!\perp (Y,W) \mid Z
\quad &\Leftrightarrow \quad 
X \perp\!\!\!\perp V \mid Z \\
&\Leftrightarrow \quad
V \perp\!\!\!\perp X \mid Z \quad &\text{Symmetry} \\
&\Rightarrow \quad
h(V) \perp\!\!\!\perp X \mid Z \quad &\text{Decomposition} \\
&\Leftrightarrow \quad
Y \perp\!\!\!\perp X \mid Z \\
&\Leftrightarrow \quad
X \perp\!\!\!\perp Y \mid Z \quad &\text{Symmetry}
\end{align}
</math>