Exercise 3.9: Conditional Mutual Information

Result $W$ as a function
of $X$, $Y$, $Z$

We assume statistically independent random variables $X$, $Y$ and $Z$ with the following properties:

$$X \in \{1,\ 2 \} \hspace{0.05cm},\hspace{0.35cm} Y \in \{1,\ 2 \} \hspace{0.05cm},\hspace{0.35cm} Z \in \{1,\ 2 \} \hspace{0.05cm},\hspace{0.35cm} P_X(X) = P_Y(Y) = \big [ 1/2, \ 1/2 \big ]\hspace{0.05cm},\hspace{0.35cm}P_Z(Z) = \big [ p, \ 1-p \big ].$$

From $X$, $Y$ and $Z$ we form the new random variable $W = (X+Y) \cdot Z$.

It is obvious that there are statistical dependencies between $X$ and $W$ ⇒ mutual information $I(X; W) ≠ 0$.
Furthermore, $I(Y; W) ≠ 0$ as well as $I(Z; W) ≠ 0$ will also apply, but this will not be discussed in detail in this exercise.

Three different definitions of mutual information are used in this exercise:

the conventional mutual information zwischen $X$ and $W$:

$$I(X;W) = H(X) - H(X|\hspace{0.05cm}W) \hspace{0.05cm},$$

the conditional mutual information between $X$ and $W$ with a given fixed value $Z = z$:

$$I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = z) = H(X\hspace{0.05cm}|\hspace{0.05cm} Z = z) - H(X|\hspace{0.05cm}W ,\hspace{0.05cm} Z = z) \hspace{0.05cm},$$

the conditional mutual information between $X$ and $W$ for a given random variable $Z$:

$$I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z ) = H(X\hspace{0.05cm}|\hspace{0.05cm} Z ) - H(X|\hspace{0.05cm}W \hspace{0.05cm} Z ) \hspace{0.05cm}.$$

The relationship between the last two definitions is:

$$I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z ) = \sum_{z \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp} (P_{Z})} \hspace{-0.2cm} P_Z(z) \cdot I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = z)\hspace{0.05cm}.$$

Hints:

The exercise belongs to the chapter Different entropies of two-dimensional random variables.
In particular, reference is made to the page Conditional mutual information.

Questions

$ I(X; W | Z = 1) \ = \ $

$\ \rm bit$

$ I(X; W | Z = 2) \ = \ $

$\ \rm bit$

$p = 1/2\text{:} \ \ \ I(X; W | Z) \ = \ $

$\ \rm bit$

$p = 3/4\text{:} \ \ \ I(X; W | Z) \ = \ $

$\ \rm bit$

$I(X; W) \ = \ $

$\ \rm bit$

Solution

Two-dimensional probability mass functions for $Z = 1$

(1) The upper graph is valid for $Z = 1$ ⇒ $W = X + Y$.

Under the conditions $P_X(X) = \big [1/2, \ 1/2 \big]$ as well as $P_Y(Y) = \big [1/2, \ 1/2 \big]$ the joint probabilities $P_{ XW|Z=1 }(X, W)$ thus result according to the right graph (grey background).

Thus the following applies to the mutual information under the fixed condition $Z = 1$:

$$I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = 1) \hspace{-0.05cm} = \hspace{-1.1cm}\sum_{(x,w) \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp} (P_{XW}\hspace{0.01cm}|\hspace{0.01cm} Z\hspace{-0.03cm} =\hspace{-0.03cm} 1)} \hspace{-1.1cm} P_{XW\hspace{0.01cm}|\hspace{0.01cm} Z\hspace{-0.03cm} =\hspace{-0.03cm} 1} (x,w) \cdot {\rm log}_2 \hspace{0.1cm} \frac{P_{XW\hspace{0.01cm}|\hspace{0.01cm} Z\hspace{-0.03cm} =\hspace{-0.03cm} 1} (x,w) }{P_X(x) \cdot P_{W\hspace{0.01cm}|\hspace{0.01cm} Z\hspace{-0.03cm} =\hspace{-0.03cm} 1} (w) }$$

$$I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = 1) = 2 \cdot \frac{1}{4} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/4}{1/2 \cdot 1/4} + 2 \cdot \frac{1}{4} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/4}{1/2 \cdot 1/2} $$

$$\Rightarrow \hspace{0.3cm} I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = 1) \hspace{0.15cm} \underline {=0.5\,{\rm (bit)}} \hspace{0.05cm}.$$

The first term summarises the two horizontally shaded fields in the graph, the second term the vertically shaded fields.
The second term do not contribute because of $\log_2 (1) = 0$ .

Two-dimensional probability mass functions for $Z = 2$

(2) For $Z = 2$, $W = \{4,\ 6,\ 8\}$ is valid, but nothing changes with respect to the probability functions compared to subtask (1).

Consequently, the same conditional mutual information is obtained:

$$I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = 2) = I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = 1) \hspace{0.15cm} \underline {=0.5\,{\rm (bit)}} \hspace{0.05cm}.$$

(3) The equation is for $Z = \{1,\ 2\}$ with ${\rm Pr}(Z = 1) =p$ and ${\rm Pr}(Z = 2) =1-p$:

$$I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z) = p \cdot I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = 1) + (1-p) \cdot I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z = 2)\hspace{0.15cm} \underline {=0.5\,{\rm (bit)}} \hspace{0.05cm}.$$

It is considered that according to subtasks (1) and (2) the conditional mutual information for given $Z = 1$ and given $Z = 2$ are equal.
Thus $I(X; W|Z)$, i.e. under the condition of a stochastic random variable $Z = \{1,\ 2\}$ with $P_Z(Z) = \big [p, \ 1 – p\big ]$ is independent of $p$.
In particular, the result is also valid for $\underline{p = 1/2}$ and $\underline{p = 3/4}$.

To calculate the joint probability for $XW$

(4) The joint probability $P_{ XW }$ depends on the $Z$–probabilites $p$ and $1 – p$ .

For $Pr(Z = 1) = Pr(Z = 2) = 1/2$ the scheme sketched on the right results.
Again, only the two horizontally shaded fields contribute to the mutual information:

$$ I(X;W) = 2 \cdot \frac{1}{8} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/8}{1/2 \cdot 1/8} \hspace{0.15cm} \underline {=0.25\,{\rm (bit)}} \hspace{0.35cm} < \hspace{0.35cm} I(X;W \hspace{0.05cm}|\hspace{0.05cm} Z) \hspace{0.05cm}.$$

The result $I(X; W|Z) > I(X; W)$ is true for this example, but also for many other applications:

If I know $Z$, I know more about the 2D random variable $XW$ than without this knowledge..
However, one must not generalize this result:

Sometimes $I(X; W) > I(X; W|Z)$, actually applies, as in Example 4 in the theory section.