Aufgaben:Exercise 3.5Z: Kullback-Leibler Distance again

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Determined probability mass functions

The probability mass function is:

$$P_X(X) = \big[\hspace{0.03cm}0.25\hspace{0.03cm}, \hspace{0.15cm} 0.25\hspace{0.15cm},\hspace{0.15cm} 0.25 \hspace{0.03cm}, \hspace{0.15cm} 0.25\hspace{0.03cm}\big]\hspace{0.05cm}.$$

The random variable  $X$  is thus characterised by

• the symbol set size  $M=4$,
• equal probabilities  $P_X(1) = P_X(2) = P_X(3) = P_X(4) = 1/4$ .

The random variable  $Y$  is always an approximation for  $X$:

• It was obtained by simulation from a uniform distribution, whereby only  $N$  random numbers were evaluated in each case.  This means:  
• $P_Y(1)$, ... , $P_Y(4)$  are not probabilities in the conventional sense.  Rather, they describe  relative frequencies.

The result of the sixth test series  (with  $N=1000)$  is thus summarised by the following probability function:

$$P_Y(X) = \big [\hspace{0.05cm}0.225\hspace{0.15cm}, \hspace{0.05cm} 0.253\hspace{0.05cm},\hspace{0.15cm} 0.250 \hspace{0.05cm}, \hspace{0.15cm} 0.272\hspace{0.05cm}\big] \hspace{0.05cm}.$$

This notation already takes into account that the random variables  $X$  and  $Y$  are based on the same alphabet  $X = \{1,\ 2,\ 3,\ 4\}$.

With these preconditions, the  informational divergence  between the two probability functions  $P_X(.)$  and  $P_Y(.)$ :

$$D(P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) = {\rm E}_X \hspace{-0.1cm}\left [ {\rm log}_2 \hspace{0.1cm} \frac{P_X(X)}{P_Y(X)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{M} P_X(\mu) \cdot {\rm log}_2 \hspace{0.1cm} \frac{P_X(\mu)}{P_Y(\mu)} \hspace{0.05cm}.$$

One calls  $D( P_X\hspace{0.05cm} || \hspace{0.05cm}P_Y)$  the (first)  Kullback-Leibler distance.

• This is a measure of the similarity between the two probability mass functions  $P_X(.)$  and  $P_Y(.)$.
• The expected value formation occurs here with regard to the  (actually equally distributed)  random variable  $X$.  This is indicated by the nomenclature  ${\rm E}_X\big[.\big]$.

A second form of Kullback-Leibler distance results from the formation of expected values with respect to the random variable  $Y$   ⇒   ${\rm E}_Y\big [.\big ]$:

$$D(P_Y \hspace{0.05cm}|| \hspace{0.05cm} P_X) = {\rm E}_Y \hspace{-0.1cm} \left [ {\rm log}_2 \hspace{0.1cm} \frac{P_Y(X)}{P_X(X)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^M P_Y(\mu) \cdot {\rm log}_2 \hspace{0.1cm} \frac{P_Y(\mu)}{P_X(\mu)} \hspace{0.05cm}.$$

Hints:

• The exercise belongs to the chapter  Some preliminary remarks on two-dimensional random variables.
• In particular, reference is made to the page  Relative entropy – Kullback-Leibler distance.
• The entropy  $H(Y)$  and the Kullback-Leibler distance  $D( P_X \hspace{0.05cm}|| \hspace{0.05cm}P_Y)$  in the above graph are to be understood in  "bit".
• The fields marked with  "???"  in the graph are to be completed by you in this task.

Questions

1

What is the entropy of the random variable  $X$ ?

$H(X)\ = \ $ $\ \rm bit$

2

What are the entropies of the random variables  $Y$  $($approximations for  $X)$?

$N=10^3\text{:} \hspace{0.5cm} H(Y) \ = \ $ $\ \rm bit$

$N=10^2\text{:} \hspace{0.5cm} H(Y) \ = \ $ $\ \rm bit$

$N=10^1\text{:} \hspace{0.5cm} H(Y) \ = \ $ $\ \rm bit$

3

Calculate the following Kullback-Leibler distances.

$N=10^3\text{:} \hspace{0.5cm} D( P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) \ = \ $ $\ \rm bit$

$N=10^2\text{:} \hspace{0.5cm} D( P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) \ = \ $ $\ \rm bit$

$N=10^1\text{:} \hspace{0.5cm} D( P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) \ = \ $ $\ \rm bit$

4

Does  $D(P_Y\hspace{0.05cm}|| \hspace{0.05cm} P_X)$  give exactly the same result in each case?

	Yes.
	No.

5

Which statements are true for the Kullback-Leibler distances with  $N = 4$?

	$D(P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) = 0$  is true.
	$D(P_X\hspace{0.05cm}|| \hspace{0.05cm} P_Y) = 0.5 \ \rm bit$  is true.
	$D(P_X\hspace{0.05cm}|| \hspace{0.05cm} P_Y)$  is infinitely large.
	$D(P_Y\hspace{0.05cm}|| \hspace{0.05cm} P_X) = 0$  holds.
	$D(P_Y\hspace{0.05cm}|| \hspace{0.05cm} P_X) = 0.5 \ \rm bit$  holds.
	$D(P_Y\hspace{0.05cm}|| \hspace{0.05cm} P_X)$  is infinitely large.

6

Do both  $H(Y)$  and  $D(P_X\hspace{0.05cm}|| \hspace{0.05cm} P_Y)$  change monotonically with  $N$?

	Yes,
	No.

Solution

Solution

(1)  With equal probabilities, with  $M = 4$:

$$H(X) = {\rm log}_2 \hspace{0.1cm} M \hspace{0.15cm} \underline {= 2\,{\rm (bit)}} \hspace{0.05cm}.$$

(2)  The probabilities for the empirically determined random variables  $Y$  generally  (not always!)  deviate from the uniform distribution the more the parameter  $N$  is smaller. 

One obtains for

• $N = 1000 \ \ \Rightarrow \ \ P_Y(Y) = \big [0.225, \ 0.253, \ 0.250, \ 0.272 \big ]$:

$$H(Y) = 0.225 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{0.225} + 0.253 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{0.253} + 0.250 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{0.250} + 0.272 \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{0.272} \hspace{0.15cm} \underline {= 1.9968\ {\rm (bit)}} \hspace{0.05cm},$$

• $N = 100 \ \ \Rightarrow \ \ P_Y(Y) = \big[0.24, \ 0.16, \ 0.30, \ 0.30\big]$:

$$H(Y) = \hspace{0.05cm}\text{...} \hspace{0.15cm} \underline {= 1.9410\ {\rm (bit)}} \hspace{0.05cm},$$

• $N = 10 \ \ \Rightarrow \ \ P_Y(Y) = \big[0.5, \ 0.1, \ 0.3, \ 0.1 \big]$:

$$H(Y) = \hspace{0.05cm}\text{...} \hspace{0.15cm} \underline {= 1.6855\ {\rm (bit)}} \hspace{0.05cm}.$$

(3)  The equation for the Kullback-Leibler distance we are looking for is:

$$D(P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) = \sum_{\mu = 1}^{4} P_X(\mu) \cdot {\rm log}_2 \hspace{0.1cm} \frac{P_X(\mu)}{P_Y(\mu)} = \frac{1/4}{{\rm lg} \hspace{0.1cm}(2)} \cdot \left [ {\rm lg} \hspace{0.1cm} \frac{0.25}{P_Y(1)} + \frac{0.25}{P_Y(2)} + \frac{0.25}{P_Y(3)} + \frac{0.25}{P_Y(4)} \right ] $$

$$\Rightarrow \hspace{0.3cm} D(P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) = \frac{1}{4 \cdot {\rm lg} \hspace{0.1cm}(2)} \cdot \left [ {\rm lg} \hspace{0.1cm} \frac{0.25^4}{P_Y(1) \cdot P_Y(2)\cdot P_Y(3)\cdot P_Y(4)} \right ] \hspace{0.05cm}.$$

The logarithm to the base  $ 2$  ⇒   $\log_2(.)$ was replaced by the logarithm to the base  $ 10$   ⇒   $\lg(.)$ for easy use of the calculator.

The following numerical results are obtained:

• for $N=1000$:

$$D(P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) = \frac{1}{4 \cdot {\rm lg} \hspace{0.1cm}(2)} \cdot \left [ {\rm lg} \hspace{0.1cm} \frac{0.25^4}{0.225 \cdot 0.253\cdot 0.250\cdot 0.272} \right ] \hspace{0.15cm} \underline {= 0.00328 \,{\rm (bit)}} \hspace{0.05cm},$$

• for $N=100$:

$$D(P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) = \frac{1}{4 \cdot {\rm lg} \hspace{0.1cm}(2)} \cdot \left [ {\rm lg} \hspace{0.1cm} \frac{0.25^4}{0.24 \cdot 0.16\cdot 0.30\cdot 0.30} \right ] \hspace{0.15cm} \underline {= 0.0442 \,{\rm (bit)}} \hspace{0.05cm},$$

• for $N=10$:

$$D(P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) = \frac{1}{4 \cdot {\rm lg} \hspace{0.1cm}(2)} \cdot \left [ {\rm lg} \hspace{0.1cm} \frac{0.25^4}{0.5 \cdot 0.1\cdot 0.3\cdot 0.1} \right ] \hspace{0.15cm} \underline {= 0.345 \,{\rm (bit)}} \hspace{0.05cm}.$$

(4)  Correct is  No, as will be shown by the example  $N = 100$ :

$$D(P_Y \hspace{0.05cm}|| \hspace{0.05cm} P_X) = \sum_{\mu = 1}^M P_Y(\mu) \cdot {\rm log}_2 \hspace{0.1cm} \frac{P_Y(\mu)}{P_X(\mu)} = 0.24\cdot {\rm log}_2 \hspace{0.1cm} \frac{0.24}{0.25} + 0.16\cdot {\rm log}_2 \hspace{0.1cm} \frac{0.16}{0.25} +2 \cdot 0.30\cdot {\rm log}_2 \hspace{0.1cm} \frac{0.30}{0.25} = 0.0407\ {\rm (bit)}\hspace{0.05cm}.$$

• In subtask  (3)  we got  $D(P_X\hspace{0.05cm}|| \hspace{0.05cm} P_Y) = 0.0442$  instead.
• This also means:   The designation „distance” is somewhat misleading.
• According to this, one would actually expect  $D(P_Y\hspace{0.05cm}|| \hspace{0.05cm} P_X)$ = $D(P_X\hspace{0.05cm}|| \hspace{0.05cm} P_Y)$ .

Probability function, entropy and Kullback-Leibler distance

(5)  Mit  $P_Y(X) = \big [0, \ 0.25, \ 0.5, \ 0.25 \big ]$  one obtains:

$$D(P_X \hspace{0.05cm}|| \hspace{0.05cm} P_Y) = 0.25\cdot {\rm log}_2 \hspace{0.1cm} \frac{0.25}{0} + 2 \cdot 0.25\cdot {\rm log}_2 \hspace{0.1cm} \frac{0.25}{0.25}+0.25\cdot {\rm log}_2 \hspace{0.1cm} \frac{0.25}{0.50}\hspace{0.05cm}.$$

• Because of the first term, the value of  $D(P_X\hspace{0.05cm}|| \hspace{0.05cm}P_Y)$  is infinitely large.
• For the second Kullback-Leibler distance holds:

$$D(P_Y \hspace{0.05cm}|| \hspace{0.05cm} P_X) = 0\cdot {\rm log}_2 \hspace{0.1cm} \frac{0}{0.25} + 2 \cdot 0.25\cdot {\rm log}_2 \hspace{0.1cm} \frac{0.25}{0.25}+ 0.50\cdot {\rm log}_2 \hspace{0.1cm} \frac{0.5}{0.25} \hspace{0.05cm}.$$

• After looking at the limits, one can see that the first term yields the result  $0$ .  The second term also yields zero, and one obtains as the final result:

$$D(P_Y \hspace{0.05cm}|| \hspace{0.05cm} P_X) = 0.50\cdot {\rm log}_2 \hspace{0.1cm} (2) \hspace{0.15cm} \underline {= 0.5\,{\rm (bit)}} \hspace{0.05cm}.$$

  Statements 3 and 5 are therefore correct:

• From this extreme example it is clear that  $D(P_Y\hspace{0.05cm}|| \hspace{0.05cm} P_X)$  is always different from  $D(P_X\hspace{0.05cm}|| \hspace{0.05cm} P_Y)$ .
• Only for the special case  $P_Y \equiv P_X$  are both Kullback-Leibler distances equal, namely zero.
• The adjacent table shows the complete result of this task.

(6)  Correct is again  No.  Although the tendency is clear:   the larger  $N$  is,

• the more  $H(Y)$  approaches in principle the final value  $H(X) = 2 \ \rm bit$ .
• the smaller the distances  $D(P_X\hspace{0.05cm}|| \hspace{0.05cm} P_Y)$  and  $D(P_Y\hspace{0.05cm}|| \hspace{0.05cm} P_X)$ become.

However, one can also see from the table that there are exceptions:

• The entropy  $H(Y)$  is smaller for  $N = 1000$  than for  $N = 400$.
• The distance  $D(P_X\hspace{0.05cm}|| \hspace{0.05cm}P_Y)$  is greater for  $N = 1000$  than for  $N = 400$.
• The reason for this is that the empirical experiment documented here with  $N = 400$  was more likely to lead to a uniform distribution than the experiment with  $N = 1000$.
• If, on the other hand, one were to start a very (infinitely) large number of experiments with  $N = 400$  and  $N = 1000$  and average over all of them, the actually expected monotonic course would actually result.