Mutual Information

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Sources:

  1. Elements of Information Theory
  2. An Introduction to Single-User Information Theory

Definition

Given two random variables \(X\) and \(Y\), we want to define a measure of the information that \(Y\) provides about \(X\) when \(Y\) is observed, but \(X\) is not. We call this measure mutual information, which is defined as: \[ I(X ; Y) \triangleq H(X)-H(X \mid Y) \]

It can be expressed in terms relative entropy between their joint distribution \(p_{X, Y}\) and the product of their marginal distributions \(p_X \cdot p_Y\) \[ \begin{aligned} I(X ; Y) & =\sum_{x, y} p(x, y) \log \frac{p_{X, Y}(x, y)}{p_X(x) \cdot p_Y(y)} \\ & =D\left(p_{X, Y} \| p_X \cdot p_Y\right) . \end{aligned} \]

Properties of Mutual Information

Basics

  1. \(I(X ; Y)=\sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} P_{X, Y}(x, y) \log _2 \frac{P_{X, Y}(x, y)}{P_X(x) P_Y(y)}\).

  2. \(I(X ; Y)=I(Y ; X)=H(Y)-H(Y \mid X)\).

  3. \(I(X ; Y)=H(X)+H(Y)-H(X, Y)\).

  4. \(I(X ; Y) \leq H(X)\) with equality holding iff \(X\) is a function of \(Y\) (i.e., \(X=f(Y)\) for some function \(f(\cdot))\).

  5. \(I(X ; Y) \geq 0\) with equality holding iff \(X\) and \(Y\) are independent.

  6. \(I(X ; Y) \leq \min \left\{\log _2|\mathcal{X}|, \log _2|\mathcal{Y}|\right\}\).

Proof:

Properties 1, 2, 3, and 4 follow immediately from the definition.

Property 5 is a direct consequence of \(D(p \| q) \geq 0\), since \(I(X:Y) = D\left(p_{X, Y} \| p_X \cdot p_Y\right)\) .

Property 6 holds iff \(I(X ; Y) \leq \log _2|\mathcal{X}|\) and \(I(X ; Y) \leq \log _2|\mathcal{Y}|\).

To show the first inequality, we write \(I(X ; Y)=H(X)-\) \(H(X \mid Y)\), use the fact that \(H(X \mid Y)\) is nonnegative and apply Theorem: \(H(X) \le |\mathcal X|\). A similar proof can be used to show that \(I(X ; Y) \leq \log _2|\mathcal{Y}|\).

The relationships between \(H(X), H(Y), H(X, Y), H(X \mid Y), H(Y \mid X)\), and \(I(X ; Y)\) can be illustrated by the Venn diagram in the above figure.

\(I(X:Y) = I(Y:X)\)

Expanding \(H(X)-H(X \mid Y)\), we have: \[ \begin{aligned} H(X)-H(X \mid Y) & =\mathbb{E}\left[\log \frac{1}{p(X)}\right]-\mathbb{E}\left[\log \frac{1}{p(X \mid Y)}\right] \\ & =\mathbb{E}\left[\log \frac{p(X \mid Y)}{p(X)}\right] \\ & =\mathbb{E}\left[\log \frac{p(X \mid Y) p(Y)}{p(X) p(Y)}\right] \\ & =\mathbb{E}\left[\log \frac{p(X, Y)}{p(X) p(Y)}\right] \\ & =H(Y)-H(Y \mid X) \end{aligned} \]

Then \[ I(X ; Y) \triangleq H(X)-H(X \mid Y)=H(Y)-H(Y \mid X) \]

So mutual information is symmetric.

\(I(X ; X)=H(X)\)

Now let’s ask an interesting question: How much does X tell us about itself? In other words, what is I (X; X)? Using our first definition, we have: \[ I (X; X) = H (X) − H (X|X) \]

We note that \(H(X \mid X)=0\), because in the expectation, \(X\) can only take on one fixed, given value with probability 1 . Therefore, \(H(X \mid X)=\log 1=0\). Thus: \[ I(X ; X)=H(X) \] Meaning that \(X\) tells us everything about itself!

\(I(X ; Y) \geq 0\)

For any two random variables, \(X, Y\), \[ I(X ; Y) \geq 0 \] with equality if and only if \(X\) and \(Y\) are independent.

Proof:

  1. We know that \(I(X ; Y)=D(p(x, y) \| p(x) p(y))\).
  2. See property 3 of Relative Entropy, \(D(p \| q) \geq 0\) for all distributions \(p, q\) with equality holding iff \(p=q\).
  3. So \(I(X ; Y)=D(p(x, y) \| p(x) p(y)) \geq 0\), with equality if and only if \(p(x, y)=p(x) p(y)\) (i.e., \(X\) and \(Y\) are independent).

Corollary: \[ I(X ; Y \mid Z) \geq 0, \] with equality if and only if \(X\) and \(Y\) are conditionally independent given \(Z\).

Conditional Mutual Information

img

Venn diagram of information theoretic measures for three variables \(x, y\), and \(z\), represented by the lower left, lower right, and upper circles, respectively.

The conditional mutual informations \(I(x ; z \mid y), I(y ; z \mid x)\) and \(I(x ; y \mid z)\) are represented by the yellow, cyan, and magenta regions, respectively.

The conditional mutual information, denoted by \(I(X ; Y \mid Z)\), is defined as the common uncertainty between \(X\) and \(Y\) under the knowledge of \(Z\) : \[ I(X ; Y \mid Z):=H(X \mid Z)-H(X \mid Y, Z) \]

可以这么想像: \(I(X;Y)\) 就是 \(H(X)\) 和 \(H(Y)\) 的交集, 再挖掉其中 \(H(Z)\) 的部分就是\(I(X;Y|Z)\). 对应于图中粉色部分.

Joint Mutual Information

Lemma: Defining the joint mutual information between \(X\) and the pair \((Y, Z)\) by \[ I(X ; Y, Z):=H(X)-H(X \mid Y, Z), \] we have \[ I(X ; Y, Z)=I(X ; Y)+I(X ; Z \mid Y)=I(X ; Z)+I(X ; Y \mid Z) . \] \(X\) 和 \((Y,Z)\)的互信息 = \(X\) 和 \(Y\) 的互信息 + 在\(Y\) 已知的情况下 \(X\) 和 \(Z\) 的互信息.

可以这么想像: 把\(H(Y), H(Z)\) 连成一块得到\(H(Y,Z)\), \(I(X;Y,Z)\) 就是 \(H(X)\) 和 \(H(Y,Z)\) 的交集. 对应于图中黄, 灰, 粉三块区域的并集.

Proof: Without loss of generality, we only prove the first equality: \[ \begin{aligned} I(X ; Y, Z) & =H(X)-H(X \mid Y, Z) \\ & =H(X)-H(X \mid Y)+H(X \mid Y)-H(X \mid Y, Z) \\ & =I(X ; Y)+I(X ; Z \mid Y) . \end{aligned} \]

The above lemma can be read as follows: the information that \((Y, Z)\) has about \(X\) is equal to the information that \(Y\) has about \(X\) plus the information that \(Z\) has about \(X\) when \(Y\) is already known.

Properties of Entropy and Mutual Information for Multiple Random Variables

Chain rule for joint entropy

Theorem: Let \(X_1, X_2, \ldots, X_n\) be drawn according to \(P_{X^n}\left(x^n\right):=P_{X_1, \ldots, X_n}\left(x_1, \ldots, x_n\right)\), where we use the common superscript notation to denote an \(n\)-tuple: \(X^n:=\left(X_1, \ldots, X_n\right)\) and \(x^n:=\left(x_1, \ldots, x_n\right)\).

Then \[ H\left(X_1, X_2, \ldots, X_n\right)=\sum_{i=1}^n H\left(X_i \mid X_{i-1}, \ldots, X_1\right), \] where \(H\left(X_i \mid X_{i-1}, \ldots, X_1\right):=H\left(X_1\right)\) for \(i=1\). (The above chain rule can also be written as: \[ H\left(X^n\right)=\sum_{i=1}^n H\left(X_i \mid X^{i-1}\right) \] where \(X^i:=\left(X_1, \ldots, X_i\right)\).)

For example, for three random variables \(X\), \(Y\) , and \(Z\), \[ \begin{aligned} H(X, Y, Z) & =H(X)+H(Y, Z \mid X) \\ & =H(X)+H(Y \mid X)+H(Z \mid X, Y) \end{aligned} \]

Proof:

From chain rule for 2 r.v. , \[ H\left(X_1, X_2, \ldots, X_n\right)=H\left(X_1, X_2, \ldots, X_{n-1}\right)+H\left(X_n \mid X_{n-1}, \ldots, X_1\right) \]

Once again, applying chain rule for 2 r.v. to the first term of the right-hand side of this equation, we have \[ H\left(X_1, X_2, \ldots, X_{n-1}\right)=H\left(X_1, X_2, \ldots, X_{n-2}\right)+H\left(X_{n-1} \mid X_{n-2}, \ldots, X_1\right) \]

The desired result can then be obtained by repeatedly applying chain rule for 2 r.v. .

Chain rule for conditional entropy

Theorem: \[ H\left(X_1, X_2, \ldots, X_n \mid Y\right)=\sum_{i=1}^n H\left(X_i \mid X_{i-1}, \ldots, X_1, Y\right) . \]

Proof:

The theorem can be proved similarly to Chain Rule for Entropy(2 Variables). If \(X^n=\left(X_1, \ldots, X_n\right)\) and \(Y^m=\left(Y_1, \ldots, Y_m\right)\) are jointly distributed random vectors (of not necessarily equal lengths), then their joint mutual information is given by \[ I\left(X_1, \ldots, X_n ; Y_1, \ldots, Y_m\right):=H\left(X_1, \ldots, X_n\right)-H\left(X_1, \ldots, X_n \mid Y_1, \ldots, Y_m\right) . \]

Chain rule for mutual information

Theorem: \[ I\left(X_1, X_2, \ldots, X_n ; Y\right)=\sum_{i=1}^n I\left(X_i ; Y \mid X_{i-1}, \ldots, X_1\right), \] where \(I\left(X_i ; Y \mid X_{i-1}, \ldots, X_1\right):=I\left(X_1 ; Y\right)\) for \(i=1\).

Proof:

This can be proved by first expressing mutual information in terms of entropy and conditional entropy, and then applying the chain rules for entropy and conditional entropy.

Independence bound on entropy

Theorem: \[ H\left(X_1, X_2, \ldots, X_n\right) \leq \sum_{i=1}^n H\left(X_i\right) . \]

Equality holds iff all the \(X_i\) 's are independent of each other.[^8]

Proof:

By applying the chain rule for entropy, \[ \begin{aligned} H\left(X_1, X_2, \ldots, X_n\right) & =\sum_{i=1}^n H\left(X_i \mid X_{i-1}, \ldots, X_1\right) \\ & \leq \sum_{i=1}^n H\left(X_i\right) . \end{aligned} \]

Equality holds iff each conditional entropy is equal to its associated entropy, that iff \(X_i\) is independent of \(\left(X_{i-1}, \ldots, X_1\right)\) for all \(i\).

Bound on mutual information

Theorem: If \(\left\{\left(X_i, Y_i\right)\right\}_{i=1}^n\) is a process satisfying the conditional independence assumption \(P_{Y^n \mid X^n}=\prod_{i=1}^n P_{Y_i \mid X_i}\), then \[ I\left(X_1, \ldots, X_n ; Y_1, \ldots, Y_n\right) \leq \sum_{i=1}^n I\left(X_i ; Y_i\right), \] with equality holding iff \(\left\{X_i\right\}_{i=1}^n\) are independent.

Proof:

From the independence bound on entropy, we have \[ H\left(Y_1, \ldots, Y_n\right) \leq \sum_{i=1}^n H\left(Y_i\right) . \]

By the conditional independence assumption, we have \[ \begin{aligned} H\left(Y_1, \ldots, Y_n \mid X_1, \ldots, X_n\right) & =E\left[-\log _2 P_{Y^n \mid X^n}\left(Y^n \mid X^n\right)\right] \\ & =E\left[-\sum_{i=1}^n \log _2 P_{Y_i \mid X_i}\left(Y_i \mid X_i\right)\right] \\ & =\sum_{i=1}^n H\left(Y_i \mid X_i\right) . \end{aligned} \]

Hence, \[ \begin{aligned} I\left(X^n ; Y^n\right) & =H\left(Y^n\right)-H\left(Y^n \mid X^n\right) \\ & \leq \sum_{i=1}^n H\left(Y_i\right)-\sum_{i=1}^n H\left(Y_i \mid X_i\right) \\ & =\sum_{i=1}^n I\left(X_i ; Y_i\right), \end{aligned} \] with equality holding iff \(\left\{Y_i\right\}_{i=1}^n\) are independent, which holds iff \(\left\{X_i\right\}_{i=1}^n\) are independent.