Exercise 1.5: Binary Markov Source

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Binary Markov diagram

  Task 1.4  has shown that the calculation of the entropy for a memory-containing source can be very time-consuming.  One must then first calculate (very many) entropy approximations  $H_k$  for  $k$–tuples and only then can the source entropy be determined with the boundary transition  $k \to \infty$ :

$$H = \lim_{k \rightarrow \infty } H_k \hspace{0.05cm}.$$

Often,  $H_k$  tends only very slowly towards the limit  $H$.

The calculation process is drastically reduced if the message source has Markov properties.  The diagram shows the transition diagram for a binary Markov source with the two states (symbols)  $\rm A$  and  $\rm B$. This is determined by the two conditional Markov sources.

  • This is clearly determined by the two conditional probabilities  $p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B} = p$  and  $p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A} = q$ .
  • The other conditional probabilities  $p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}A}$  and  $p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}B}$  as well as the (unconditional) symbol probabilities  $p_{\rm A}$   and  $p_{\rm B}$  can be determined from this.


The entropy of the binary Markov chain  (with the unit "bit/symbol")  is then:

$$H = H_{\rm M} = p_{\rm AA} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}A}} + p_{\rm AB} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A}} + p_{\rm BA} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B}} + p_{\rm BB} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}B}} \hspace{0.05cm}.$$

In this equation, it should be noted that in the argument of the binary logarithm , the  conditional probabilities  $p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}A}$,  $p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A}$, ...   are to be used, while the  joint probabilities  $p_{\rm AA}$,  $p_{\rm AB}$, ...   are to be used for the weighting.

Using the first order entropy approximation,

$$H_1 = p_{\rm A} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm A}} + p_{\rm B} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm B}} \hspace{0.5cm}({\rm unit\hspace{-0.1cm}: \hspace{0.1cm}bit/symbol})\hspace{0.05cm},$$

as well as the (actual) entropy  $H = H_{\rm M}$  given above, all further entropy approximations  $(k = 2,, 3, \text{...})$  can also be given directly for a Markov source:

$$H_k = \frac{1}{k} \cdot \big [ H_{\rm 1} + (k-1) \cdot H_{\rm M} \big ] \hspace{0.05cm}.$$





Hints:

  • For the (ergodic) symbol probabilities of a first order Markov chain applies:
$$ p_{\rm A} = \frac {p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B}} { p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B} + p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A}} \hspace{0.05cm}, \hspace{0.3cm} p_{\rm B} = \frac {p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A}} { p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B} + p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A}} \hspace{0.05cm}.$$



Questions

1

Give the transition probabilities for  $p = 1/4$  and  $q = 1/2$.

$p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}A} \ = \ $

$p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}B} \ = \ $

2

What are the (unconditional) symbol probabilities?  Let  $p = 1/4$  and  $q = 1/2$ still hold.

$p_{\rm A} \ = \ $

$p_{\rm B} \ = \ $

3

Give the corresponding first-order entropy approximation.

$H_1 \ = \ $

$\ \rm bit/symbol$

4

What is the entropy  $H = H_{\rm M}$  of this Markov source with  $p = 1/4$  and  $q = 1/2$?

$H \ = \ $

$\ \rm bit/symbol$

5

Which entropy approximations  $H_k$  result from the Markov properties?

$H_2 \ = \ $

$\ \rm bit/symbol$
$H_3 \ = \ $

$\ \rm bit/symbol$
$H_4 \ = \ $

$\ \rm bit/symbol$

6

What is the entropy  $H = H_{\rm M}$  of the Markov source with  $p = 1/4$  and  $q = 3/4$?

$H \ = \ $

$\ \rm bit/symbol$


Solution

Markov diagram for the subtasks  (1), ... ,  (5)

After  $\rm A$ ,   $\rm A$  and  $\rm B$  are equally probable.  After  $\rm B$ ,   $\rm B$  occurs much more frequently than  $\rm A$ .  The following applies to the transition probabilities

$$p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}A} \hspace{0.1cm} = \hspace{0.1cm} 1 - p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A}= 1 - q \hspace{0.15cm} \underline {= 0.5} \hspace{0.05cm},$$
$$ p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}B} \hspace{0.1cm} = \hspace{0.1cm} 1 - p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B}= 1 - p \hspace{0.15cm} \underline {= 0.75} \hspace{0.05cm}.$$


(2)  According to the equations given:

$$p_{\rm A}= \frac{p}{p+q} = \frac{0.25}{0.25 + 0.50} \hspace{0.15cm} \underline {= 0.333} \hspace{0.05cm}, \hspace{0.5cm} p_{\rm B} = \frac{q}{p+q} = \frac{0.50}{0.25 + 0.50} \hspace{0.15cm} \underline {= 0.667} \hspace{0.05cm}.$$


(3)  With the probabilities calculated in the last sub-task:

$$H_{\rm 1} = H_{\rm bin}(p_{\rm A}) = 1/3 \cdot {\rm log}_2\hspace{0.01cm} (3) + 2/3 \cdot {\rm log}_2\hspace{0.01cm} (1.5) = 1.585 - 2/3\hspace{0.15cm} \underline {= 0.918 \,{\rm bit/symbol}} \hspace{0.05cm}.$$


(4)  The entropy of the Markov source is according to:

$$H = p_{\rm AA} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}A}} + p_{\rm AB} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A}} + p_{\rm BA} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B}} + p_{\rm BB} \cdot {\rm log}_2\hspace{0.1cm}\frac {1}{ p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}B}} \hspace{0.05cm}.$$
  • For the composite probabilities holds:
$$p_{\rm AA} = p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}A} \cdot p_{\rm A} = (1-q) \cdot \frac{p}{p+q} = \frac{1/2 \cdot 1/4}{3/4} = {1}/{6} \hspace{0.05cm},$$
$$ p_{\rm AB} = p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A} \cdot p_{\rm A} = q \cdot \frac{p}{p+q} = \frac{1/2 \cdot 1/4}{3/4} = {1}/{6} \hspace{0.05cm},$$
$$ p_{\rm BA} = p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B} \cdot p_{\rm B} = p \cdot \frac{q}{p+q} = p_{\rm AB} = {1}/{6} \hspace{0.05cm},$$
$$ p_{\rm BB} = p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}B} \cdot p_{\rm B} = (1-p) \cdot \frac{q}{p+q} = \frac{3/4 \cdot 1/2}{3/4} = {1}/{2} $$
$$\Rightarrow\hspace{0.3cm} H = 1/6 \cdot {\rm log}_2\hspace{0.01cm} (2) + 1/6 \cdot {\rm log}_2\hspace{0.01cm} (2) + 1/6 \cdot {\rm log}_2\hspace{0.01cm} (4) + 1/2 \cdot {\rm log}_2\hspace{0.1cm} (4/3) = 10/6 - 1/2 \cdot {\rm log}_2\hspace{0.01cm} (3) \hspace{0.15cm} \underline {= 0.875 \,{\rm bit/symbol}} \hspace{0.05cm}.$$


(5)  In general, with  $H_{\rm M} = H$  for  $k$–th entropy approximation:  

$$H_k = {1}/{k} \cdot [ H_{\rm 1} + (k-1) \cdot H_{\rm M}] \hspace{0.05cm}.$$
  • It follows that:
$$H_2 = {1}/{2} \cdot [ 0.918 + 1 \cdot 0.875] \hspace{0.15cm} \underline {= 0.897 \,{\rm bit/symbol}} \hspace{0.05cm},$$
$$ H_3 = {1}/{3} \cdot [ 0.918 + 2 \cdot 0.875] \hspace{0.15cm} \underline {= 0.889 \,{\rm bit/symbol}} \hspace{0.05cm},$$
$$ H_4 = {1}/{4} \cdot [ 0.918 + 3 \cdot 0.875] \hspace{0.15cm} \underline {= 0.886 \,{\rm bit/symbol}} \hspace{0.05cm}.$$


Markov diagram for subtask  (6)

(6)  With the new set of parameters  $(p = 1/4, q = 3/4)$  , we obtain for the symbol probabilities:

$$ p_{\rm A} = 1/4, \ p_{\rm B} = 3/4.$$
  • This special case thus leads to statistically independent symbols:
$$ p_{\rm A} = p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}A} = p_{\rm A\hspace{0.01cm}|\hspace{0.01cm}B} \hspace{0.05cm}, \hspace{0.2cm} p_{\rm B} = p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}A} = p_{\rm B\hspace{0.01cm}|\hspace{0.01cm}B} \hspace{0.05cm}.$$
  • Thus the entropy  $H$  is identical with the entropy approximation  $H_1$:
$$H = H_{\rm 1} = 1/4 \cdot {\rm log}_2\hspace{0.01cm} (4) + 3/4 \cdot {\rm log}_2\hspace{0.01cm} (4/3) = 2 - 0.75 \cdot {\rm log}_2\hspace{0.01cm} (3) \hspace{0.15cm} \underline {= 0.811 \,{\rm bit/Symbol}} \hspace{0.05cm}.$$
  • The entropy approximations  $H_2$,  $H_3$,  $H_4$,  ...  also yield the result  $0.811 \, \rm bit/symbol$.