Entropy, relative entropy and mutual information

Entropy
Chain Rules
Jensen's Inequality
- Properties
Log Sum Inequality
Data-Processing Inequality
- Sufficient Statistics
  - Statistics and Mutual Information
  - Sufficient Statistics and Compression

Entropy, relative entropy and mutual information.

Entropy

\[H(X) = -\sum_{x} p(x) \log p(x),
\]

熵非负, 且当且仅当\(X\)确定性的时候为有最小值0, 即\(P(X=x_0)=1\).

Proof:

由\(\log\)的凹性可得

\[\begin{array}{ll}
H(X)
& = -\sum_{x} p(x) \log p(x) \\
& = \sum_{x} p(x) \log \frac{1}{p(x)} \\
& \ge \log 1=0.
\end{array}
\]

Joint Entropy

\[H(X,Y) := -\mathbb{E}_{p(x, y)} [\log p(x, y)] = \sum_{x \in \mathcal{X}} \sum_{y\in \mathcal{Y}} p(x, y) \log p(x, y).
\]

Conditional Entropy

\[\begin{array}{ll}
H(Y|X)
&= - \mathbb{E}_{p(x)} [H(Y|X=x)] \\
&= - \sum_{x \in \mathcal{X}} p(x) H(Y|X=x) \\
&= - \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} p(x)p(y|x) \log p(y|x) \\
&= - \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} p(x, y) \log p(y|x).
\end{array}
\]

注意 \(H(Y|X)\) 和 \(H(Y|X=x)\) 的区别.

Chain rule

\[H(X, Y) = H(X) + H(Y|X).
\]

proof:

根据\(p(y|x)=\frac{p(x, y)}{p(x)}\)以及上面的推导可知:

\[\begin{array}{ll}
H(Y|X)
&= H(X,Y) + \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} p(x, y) \log p(x) \\
&= H(X, Y) -H(X).
\end{array}
\]

推论:

\[H(X,Y| Z) = H(X|Z) + H(Y|X, Z).
\]

\[\begin{array}{ll}
H(Y|X,Z)
&= \mathbb{E}_{x,z} [H(Y|x,z)] \\
&= -\sum_{x,z} p(x,z) p(y|x,z) \log p(y|x,z) \\
&= -\sum_{x, z} p(x, y, z) [\log p(x, y|z) - \log p(x|z)] \\
&= \mathbb{E}_{z} H(X, Y|z) - \mathbb{E}_{z} H(X|z) = H(X, Y|Z) - H(X|Z).
\end{array}
\]

Mutual Information

\[I(X;Y) = H(X) - H(X|Y) = \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}}p(x, y)\log \frac{p(x, y)}{p(x)p(y)}
\]

\(I(X; Y) = H(X) - H(X|Y) = H(Y) - H(Y|X) = I(Y;X) = H(X) + H(Y) - H(X, Y) \ge 0\)
\(I(X, X) = H(X)\)

Relative Entropy

\[D(p\|q) := \mathbb{E}_p (\log \frac{p(x)}{q(x)}) = \sum_{x\in \mathcal{X}} p(x) \log \frac{p(x)}{q(x)}.
\]

Chain Rules

Chain Rule for Entropy

设\((X_1, X_2,\ldots, X_n) \sim p(x_1, x_2, \ldots, x_n)\):

\[H(X_1, X_2, \ldots, X_n) = \sum_{i-1}^n H(X_i|X_{i-1}, \ldots, X_1).
\]

proof:

归纳法 + \(H(X, Y) = H(X) + H(Y|X)\).

Chain Rule for Mutual Information

Conditional Mutual Information

定义:

\[I(X;Y|Z) := H(X|Z) - H(X|Y,Z)= \mathbb{E}_{p(x, y,z)} \log \frac{p(X, Y| Z)}{p(X|Z)p(Y|Z)}.
\]

性质:

\[I(X_1, X_2, \ldots, X_n; Y) = \sum_{i=1}^n I(X_i;Y|X_{i-1}, \ldots, X_1).
\]

proof:

\[\begin{array}{ll}
I(X_1, X_2, \ldots, X_n; Y)
& =H(X_1, \ldots, X_n) + H(Y) - H(X_1,\ldots, X_n;Y) \\
&= H(X_1,\ldots, X_{n-1}) + H(X_n|X_1,\ldots, X_{n-1}) + H(Y) - H(X_1, \ldots, X_n;Y) \\
&= I(X_1, X_2,\ldots, X_{n-1};Y) + H(X_n|X_1,\ldots, X_{n-1}) - H(X_n|X_1, \ldots, X_{n-1};Y) \\
&= I(X_1, X_2,\ldots, X_{n-1};Y) + I(X_n;Y|X_1,\ldots, X_{n-1}). \\
\end{array}
\]

Chain Rule for Relative Entropy

定义:

\[\begin{array}{ll}
D(p(y|x)\|q(y|x))
&:= \mathbb{E}_{p(x, y)} [\log \frac{p(Y| X)}{q(Y|X)}] \\
&= \sum_x p(x) \sum_y p(y|x) \log \frac{p(y|x)}{q(y|x)}.
\end{array}
\]

性质:

\[D(p(x, y) \| q(x, y)) = D(p(x) \| q(x)) + D(p(y|x)\| q(y|x)).
\]

proof:

\[\begin{array}{ll}
D(p(x, y)\| q(x, y))
&= \sum_{x, y} p(x, y) \log \frac{p(x, y)}{q(x, y)} \\
&= \sum_{x, y} p(x, y) \log \frac{p(y|x)p(x)}{q(y|x)q(x)} \\
&= \sum_{x, y} [p(x, y) (\log \frac{p(y|x)}{q(y|x)} + \log \frac{p(x)}{q(x)})]\\
&= D(p(x)\|q(x)) + D(p(y|x)\|q(y|x)).
\end{array}
\]

补充:

\[D(p(x, y) \| q(x, y)) = D(p(y) \| q(y)) + D(p(x|y)\| q(x|y)).
\]

故, 当\(p(x) = q(x)\)的时候, 我们可以得到

\[D(p(x, y) \| q(x, y)) = D(p(y|x)\| q(y|x)) \ge D(p(y)\|q(y))
\]

\(D(p(y|x)\|q(y|x))=D(p(x, y)\| p(x)q(y|x))\)
\(D(p(x_1, x_2,\ldots, x_n)\| q(x_1, x_2,\ldots, x_m)) = \sum_{i=1}^n D(p(x_i|x_{i-1}, \ldots, x_1)\|q(x_i| x_{i-1}, \ldots, x_1))\)
\(D(p(y)\| q(y)) \le D(p(y|x)\|q(y|x))\), \(q(x)=p(x)\).

1, 2, 3的证明都可以通过上面的稍作变换得到.

Jensen's Inequality

如果\(f\)是凸函数, 则

\[\mathbb{E} [f(X)] \ge f(\mathbb{E}[X]).
\]

Properties

\(D(p\|q) \ge 0\) 当且仅当\(p=q\)取等号.
\(I(X; Y) \ge 0\)当且仅当\(X, Y\)独立取等号.
\(D(p(y|x)\|q(y|x)) \ge 0\) (根据上面的性质)，当且仅当\(p(y|x) = q(y|x)\)取等号, \(p(x) > 0\).
\(I(X; Y|Z) \ge 0\), 当且仅当\(X, Y\)条件独立.
\(H(X|Y)\le H(X)\), 当且仅当\(X, Y\)独立等号成立.
\(H(X_1, X_2, \ldots, X_n)\le \sum_{i=1}^n H(X_i)\), 当且仅当所有变量独立等号成立.

Log Sum Inequality

\(D(p\|q)\) 关于\((p, q)\)为凸函数, 即\(\forall 0\le \lambda \le 1\):
\[D(\lambda p_1 + (1-\lambda)p_2\| \lambda q_1 + (1-\lambda)q_2) \le \lambda D(p_1\|q_1) + (1-\lambda)D(p_2 \| q_2).
\]

此部分的证明, 一方面可以通过\(p\log\frac{p}{q}\)的凸性得到, 更有趣的证明是, 构造一个新的联合分布

\[p(x,c) = p_1 \cdot \lambda + p_2 \cdot (1- \lambda), q(x, c) = q_1 \cdot \lambda + q_2 \cdot (1-\lambda).
\]

即

\[p(x|c=0)=p_1, p(x|c=1)=p_2, q(x|c=0)=q_1, q(x|c=2)=q_2, \\
p(c=0)=q(c=0)=\lambda, p(c=1) = q(c=1) = 1-\lambda.
\]

并注意到\(D(p(y)\| q(y)) \le D(p(y|x)\|q(y|x))\).

\(H(X) = -\sum_{x \in \mathcal{X}} p(x) \log p(x)\)是关于\(p\)的凹函数.
\(I(X, Y) = \sum_{x, y} p(y|x)p(x) \log \frac{p(y|x)}{p(y)}\), 当固定\(p(y|x)\)的时候是关于\(p(x)\)的凹函数, 当固定\(p(x)\)的时候, 是关于\(p(y|x)\)的凸函数.

仅仅证明后半部分, 任给\(p_1(y|x), p_2(y|x)\), 由于\(p(x)\)固定, 故\(\forall 0 \le \lambda \le 1\):

\[p(x, y) := \lambda p_1(x, y) + (1-\lambda) p_2(x, y) = [\lambda p_1(y|x) + (1-\lambda) p_2(y|x)]p(x) \\
p(y): = \sum_x p(x, y) = \lambda \sum_x p_1(x, y) + (1-\lambda) \sum_{x} p_2(x, y) \\
q(x, y):= p(x)p(y) = \sum_x p(x, y) = \lambda p(x) \sum_x p_1(x, y) + (1-\lambda) p(x)\sum_{x} p_2(x, y) =: \lambda q_1(x, y) + (1-\lambda)q_2(x, y).\\
\]

又

\[I(X, Y) = D(p(x, y)\| p(x)p(y))=D(p(x, y)\| q(x,y)),
\]

因为KL散度关于\((p, q)\)是凸函数, 所以\(I\)关于\(p(y|x)\)如此.

Data-Processing Inequality

数据\(X \rightarrow Y \rightarrow Z\), 即\(P(X, Y,Z) = P(X)P(Y|X)P(Z|Y)\) 比如\(Y=f(X), Z = g(Y)\).

\[I(Y, Z;X) = I(X;Y) + I(X;Z|Y)= I(X;Z) + I(X;Y|Z),
\]

又

\[I(X;Z|Y) = \sum_{x, y, z} p(x, y, z) \log \frac{p(x,z|y)}{p(x|y)p(z|y)} = \sum_{x,y,z}p(x,y,z) \log 1 = 0. \\
I(X;Y|Z) = \sum_{x,y,z} p(x,y,z) \log \frac{p(x|y)}{p(x|z)}\ge 0.
\]

故

\[I(X;Z) \le I(X;Y) \\
I(X;Y|Z) \le I(X;Y).
\]

Sufficient Statistics

Statistics and Mutual Information

一族概率分布\(\{f_{\theta(x)}\}\)
\(X \sim f_{\theta}(x)\), \(T(X)\)为其统计量, 则

\[\theta \rightarrow X \rightarrow T(X)
\]
故

\[I(\theta;X) \ge I(\theta;T(X))
\]

Sufficient Statistics and Compression

充分统计量定义: 一个函数\(T(X)\)被称之为一族概率分布\(\{f_{\theta}(x)\}\)的充分统计量, 如果给定\(T(X)=t\)时\(X\)的条件分布与\(\theta\)无关, 即

\[f_{\theta}(x) = f(x|t) f_{\theta}(t) \Rightarrow \theta \rightarrow T(X) \rightarrow X \Rightarrow I(\theta;T(X)) \ge I(\theta;X).
\]

此时, \(I(\theta;T(X))= I(\theta;X)\).

最小充分统计量定义: 如果一个充分统计量\(T(X)\)与其余的一切关于\(\{f_{\theta}(x)\}\)的充分统计量\(U(X)\)满足

\[\theta \rightarrow T(X) \rightarrow U(X) \rightarrow X.
\]