Introduction to PGM
这篇文章作为 An Introduction to PGM, Jodan 的简单总结.
条件独立(Conditional Independance)
基本定义
条件独立的定义:
对于随机变量\(X, Y, Z\), 在给定Z的情况下, X和Y条件独立, 记作 \(X \perp Y \mid Z\), 当且仅当:
\[f_{X, Y \mid Z}(x, y \mid z) = f_{X \mid Z}(x \mid z)f_{Y \mid Z}(y \mid z)\]
条件独立的等价形式:
\[X \perp Y \mid Z\]
当且仅当:
\[f_{X \mid Y,Z}(x \mid y,z) = f_{X \mid Z}(x \mid z)\]
证明如下
如果
\[\begin{align}
f(x, y \mid z) = f(x \mid z)f(y \mid z) & \Leftrightarrow \frac{f(x, y, z)}{f(z)} = \frac{f(x, y)}{f(z)} \frac{f(y, z)}{f(z)} \\
& \Leftrightarrow \frac{f(x, y, z)}{f(y, z)} = \frac{f(x, y)}{f(z)} \\
& \Leftrightarrow f(x \mid y, z) = f(x, z)
\end{align}\]
条件独立的推论
\[\begin{eqnarray}
\tag{1}
X \perp Y \mid Z & \Rightarrow & Y \perp X \mid Z \\
\tag{2}
Y \perp X \mid Z & \Rightarrow & Y \perp h(X) \mid Z \\
\tag{3}
Y \perp X \mid Z & \Rightarrow & Y \perp X \mid \{Z, h(X)\} \Leftrightarrow Y \perp \{X, h(X)\} \mid Z \\
\tag{4}
Y \perp X \mid Z & \ \ and \ \ & W \perp X\ \mid\ \{Y, Z\} \Rightarrow \{Y, W\} \perp X \mid Z \\
\tag{5}
Y \perp X \mid Z & and & Z \perp X \mid Y \Rightarrow \{Y, Z\} \perp X
\end{eqnarray}\]
证明2
TODO
证明3
TODO
证明4
TODO
证明5
TODO
(3) 中 \(Y \perp X \mid \{Z, h(X)\} \Leftrightarrow Y \perp \{X, h(X)\} \mid Z\) 可采用条件独立的等价形式进行证明