Exercise 1.1Z: Binary Entropy Function

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Binary entropy function
in "bits" und "nats"

We consider a sequence of binary random variables with the symbol set  $\{ \rm A, \ B \}$   ⇒   $M = 2$.  Let the probabilities of occurrence of the two symbols be  $p_{\rm A }= p$  and  $p_{\rm B } = 1 - p$.

The individual sequence elements are statistically independent.  The entropy of this message source is equally valid:

$$H_{\rm bin}(p) \hspace{0.1cm} = \hspace{0.1cm} p \cdot {\rm ld}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p\hspace{0.1cm}} + (1-p) \cdot {\rm ld}\hspace{0.1cm}\frac{1}{1-p}\hspace{0.15cm}{\rm in \hspace{0.15cm} \big [bit \big ]}\hspace{0.05cm},$$
$$ H'_{\rm bin}(p) \hspace{0.1cm} = \hspace{0.1cm} p \cdot {\rm ln}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p\hspace{0.1cm}} + (1-p) \cdot {\rm ln}\hspace{0.1cm}\frac{1}{1-p}\hspace{0.15cm}{\rm in \hspace{0.15cm} \big [nat\big ]}\hspace{0.05cm}.$$

In these equations, the shorthand terms used are:

  • the  "natural"  logarithm   ⇒   $ {\ln} \hspace{0.09cm} p = \log_{\rm e} \hspace{0.05cm} p$,
  • the  "binary"  logarithm   ⇒   ${\rm ld} \hspace{0.09cm} p = \log_2 \hspace{0.05cm} p$.


The plot shows the binary entropy function as a function of the parameter  $p$, assuming  $0 ≤ p ≤ 1$ .

In subtasks  (5)  and  (6)  the relative error is to be determined if the symbol probability  $p$  was determined by simulation  $($i.e., as a relative frequency  $h)$,  resulting in  $h = 0.9 \cdot p$  by mistake.  The relative error is then given as follows:

$$\varepsilon_{H} = \frac{H_{\rm bin}(h)- H_{\rm bin}(p)}{H_{\rm bin}(p)}\hspace{0.05cm}.$$




Hint:


Questions

1

How are  $H_{\rm bin}(p)$  related to the unit "bit" and  $H'_{\rm bin}(p)$  related to the unit "nat"?

$H_{\rm bin}(p)$  and  $H'_{\rm bin}(p)$  differ by a factor.
It holds that  $H'_{\rm bin}(p) = H_{\rm bin}(\ln \ p)$.
It holds that  $H'_{\rm bin}(p) = 1 + H_{\rm bin}(2 p)$.

2

Show that the maximum of the binary entropy function is obtained for  $p = 0.5$ .  What is  $H_\text{bin}(p = 0.5)$?

$H_\text{bin}(p = 0.5) \ = \ $

$\ \rm bit$

3

Calculate the binary entropy value for  $p = 0.05$.

$H_\text{bin}(p = 0.05) \ = \ $

$\ \rm bit$

4

Enter the larger of the two  $p$–values given by the equation   $H_\text{bin}(p)= 0.5 \ \rm bit$ .

$p \ = \ $

5

Due to insufficient simulation,  $p = 0.5$  was determined  $10\%$ too low.  What is the percentage error with respect to the entropy?

$p = 0.45\ \ {\rm instead of}\ \ p=0.5\hspace{-0.1cm}:\ \ \varepsilon_H \ = \ $

$\ \rm \%$

6

Due to insufficient simulation,  $p = 0.05$  was determined  $10\%$  too low.  What is the percentage error with respect to the entropy here?

$p = 0.045\ \ {\rm statt}\ \ p=0.05\hspace{-0.1cm}:\ \ \varepsilon_H \ = \ $

$\ \rm \%$


Solution

(1)  The first suggested solution is correct. The other two specifications do not make sense.

  • The entropy function  $H'_{\rm bin}(p)$  is according to the specification:
$$H'_{\rm bin}(p) \hspace{0.1cm} = \hspace{0.1cm} p \cdot {\rm ln}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p\hspace{0.1cm}} + (1-p) \cdot {\rm ln}\hspace{0.1cm}\frac{1}{1-p} = {\rm ln}\hspace{0.1cm}2 \cdot \left [ p \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p\hspace{0.1cm}} + (1-p) \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{1-p}\right ]$$
$$\Rightarrow \hspace{0.3cm} H'_{\rm bin}(p) \hspace{0.15cm}{\rm (in \hspace{0.15cm} nat)}= {\rm ln}\hspace{0.1cm}2 \cdot H_{\rm bin}(p) \hspace{0.15cm}{\rm (in \hspace{0.15cm} bit)} = 0.693\cdot H_{\rm bin}(p)\hspace{0.05cm}.$$


(2)  The optimisation condition is  ${\rm d}H_{\rm bin}(p)/{\rm d}p = 0$  resp.

$$\frac{{\rm d}H'_{\rm bin}(p)}{{\rm d}p} \stackrel{!}{=} 0 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \frac{\rm d}{{\rm d}p} \big [ - p \cdot {\rm ln}\hspace{0.1cm}p - (1-p) \cdot {\rm ln}\hspace{0.1cm}({1-p})\big ] \stackrel{!}{=} 0$$
$$\Rightarrow \hspace{0.3cm} - {\rm ln}\hspace{0.1cm}p - p \cdot \frac {1}{p}+ {\rm ln}\hspace{0.1cm}(1-p) + (1-p)\cdot \frac {1}{1- p}\stackrel{!}{=} 0$$
$$\Rightarrow \hspace{0.3cm} {\rm ln}\hspace{0.1cm}\frac {1-p}{p}= 0 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}\frac {1-p}{p}= 1 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \underline { p = 0.5}\hspace{0.05cm}.$$
  • The entropy values for  $p = 0.5$  are thus:
$$H'_{\rm bin}(p = 0.5) \hspace{0.1cm} = \hspace{0.1cm} -2 \cdot 0.5 \cdot {\rm ln}\hspace{0.1cm}0.5 = {\rm ln}\hspace{0.1cm}2 = 0.693 \, {\rm nat}\hspace{0.05cm},$$
$$ H_{\rm bin}(p = 0.5) \hspace{0.1cm} = \hspace{0.1cm} -2 \cdot 0.5 \cdot {\rm ld}\hspace{0.1cm}0.5 = {\rm log_2}\hspace{0.1cm}2 \hspace{0.15cm}\underline {= 1 \, {\rm bit}}\hspace{0.05cm}.$$


(3)  For  $p = 5\%$  we get:

$$H_{\rm bin}(p = 0.05) \hspace{0.1cm} = \hspace{0.1cm} 0.05 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.05}+ 0.95 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.95}= \frac{1}{0.693} \cdot \big [ 0.05 \cdot {\rm ln}\hspace{0.1cm}20+ 0.95 \cdot {\rm ln}\hspace{0.1cm}1.053\big ] \hspace{0.15cm}\underline {\approx 0.286 \, {\rm bit}}\hspace{0.05cm}.$$


(4)  This sub-task cannot be solved in closed form, but by "trial and error".

  • A solution gives the result:
$$H_{\rm bin}(p = 0.10) = 0.469 \, {\rm bit}\hspace{0.05cm},\hspace{0.2cm}H_{\rm bin}(p = 0.12) = 0.529 \, {\rm bit}\hspace{0.05cm},\hspace{0.2cm} H_{\rm bin}(p = 0.11) \approx 0.5 \, {\rm bit} \hspace{0.3cm} \Rightarrow \hspace{0.3cm}p_1 \approx 0.11\hspace{0.05cm}. $$
  • The second solution results from the symmetry of   $H_{\rm bin}(p)$  to  $p_2 = 1 -p_1 \hspace{0.15cm}\underline{= 0.89}$.



(5)  With  $p = 0.45$  one obtains  $H_{\rm bin}(p) = 0.993\hspace{0.05cm}\rm bit$.  The relative error with respect to entropy is thus

$$\varepsilon_{H} = \frac{H_{\rm bin}(p = 0.45)- H_{\rm bin}(p= 0.5)}{H_{\rm bin}(p = 0.5)}= \frac{0.993- 1}{1}\hspace{0.15cm}\underline {= -0.7 \, {\rm \%}} \hspace{0.05cm}.$$
  • The minus sign indicates that the entropy value  $H_{\rm bin}(p) = 0.993\hspace{0.05cm}\rm bit$  is too small.
  • If the simulation had yielded the too large value  $p = 0.55$ , the entropy and also the relative error would be just as large.



(6)    $H_{\rm bin}(p = 0.045) = 0.265\hspace{0.05cm}\rm bit$ is valid.

  • With the result of subtask  (3)   ⇒   $H_{\rm bin}(p = 0.05) = 0.286\hspace{0.05cm}\rm bit$  it follows for the relative error with respect to the entropy:
$$\varepsilon_{H} = \frac{H_{\rm bin}(p = 0.045)- H_{\rm bin}(p= 0.05)}{H_{\rm bin}(p = 0.05)}= \frac{0.265- 0.286}{0.286}\hspace{0.15cm}\underline {= -7.3 \, {\rm \%}} \hspace{0.05cm}.$$
  • The result shows:
  1.  An incorrect determination of the symbol probabilities by  $10\%$  is much more noticeable for  $p = 0.05$  due to the steeper  $H_{\rm bin}(p)$ course than for  $p = 0.5$.
  2.  A too large probability  $p = 0.055$  would have led to  $H_{\rm bin}(p = 0.055) = 0.307\hspace{0.05cm}\rm bit$  and thus to a distortion of  $\varepsilon_H = +7.3\%$. 
  3. In this range, the entropy curve is thus linear (with a good approximation).