Aufgaben:Exercise 3.9: Conditional Mutual Information - Revision history

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	Hints:		Hints:
	*The exercise belongs to the chapter  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen\|Different entropies of two-dimensional random variables]].		*The exercise belongs to the chapter  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen\|Different entropies of two-dimensional random variables]].
	*In particular, reference is made to the page  [[Information_Theory/~~Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen~~#Conditional_mutual_information\|Conditional mutual information]].		*In particular, reference is made to the page  [[Information_Theory/Different_Entropy_Measures_of_Two-Dimensional_Random_Variables#Conditional_mutual_information\|Conditional mutual information]].


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	[[Category:Information Theory: Exercises\|^3.2 Entropies of 2D Random Variables^]]		[[Category:Information Theory: Exercises\|^3.2 Entropies of 2D Random Variables^]]
	[[de:Aufgaben:3.~~08_Bedingte_Transinformation~~]]		[[de:Aufgaben:Aufgabe 3.9: Bedingte Transinformation]]

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	[[File:P_ID2813__Inf_A_3_8.png\|right\|frame\|Result  $W$  as a function <br>of  $X$,  $Y$,  $Z$]]		[[File:P_ID2813__Inf_A_3_8.png\|right\|frame\|Result  $W$  as a function <br>of  $X$,  $Y$,  $Z$]]
	We assume statistically independent random variables  $X$,  $Y$  and  $Z$  with the following properties:		We assume statistically independent random variables  $X$,  $Y$  and  $Z$  with the following properties:
	:$$X \in \{1,\ 2 \} \hspace{0.05cm},\hspace{0.35cm}		:$$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 ].$$
	Y \in \{1,\ 2 \} \hspace{0.05cm},\hspace{0.35cm}
	Z \in \{1,\ 2 \} \hspace{0.05cm},\hspace{0.35cm} P_X(X) = P_Y(Y) = \big [ 1/2, \ 1/2 \big ]\hspace{0.05cm},\hspace{0.35cm}P_Z(Z) = \big [ p, \ 1-p \big ].$$

	From  $X$,  $Y$  and  $Z$  we form the new random variable  $W = (X+Y) \cdot Z$.		From  $X$,  $Y$  and  $Z$  we form the new random variable  $W = (X+Y) \cdot Z$.
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	The relationship between the last two definitions is:		The relationship between the last two definitions is:
	:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z ) = \sum_{z \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp} (P_{Z})} \hspace{-0.2cm}		:$$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}.$$
	P_Z(z) \cdot I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = z)\hspace{0.05cm}.$$


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	*Thus the following applies to the mutual information under the fixed condition  $Z = 1$:		*Thus the following applies to the mutual information under the fixed condition  $Z = 1$:
	:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 1) \hspace{-0.05cm} = \hspace{-1.1cm}\sum_{(x,w) \hspace{0.1cm}\in \hspace{0.1cm}{\rm supp} (P_{XW}\hspace{0.01cm}\|\hspace{0.01cm} Z\hspace{-0.03cm} =\hspace{-0.03cm} 1)} \hspace{-1.1cm}		:$$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) }$$
	P_{XW\hspace{0.01cm}\|\hspace{0.01cm} Z\hspace{-0.03cm} =\hspace{-0.03cm} 1} (x,w) \cdot {\rm log}_2 \hspace{0.1cm} \frac{P_{XW\hspace{0.01cm}\|\hspace{0.01cm} Z\hspace{-0.03cm} =\hspace{-0.03cm} 1} (x,w) }{P_X(x) \cdot P_{W\hspace{0.01cm}\|\hspace{0.01cm} Z\hspace{-0.03cm} =\hspace{-0.03cm} 1} (w) }$$		:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 1) = 2 \cdot \frac{1}{4} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/4}{1/2 \cdot 1/4} +2 \cdot \frac{1}{4} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/4}{1/2 \cdot 1/2}$$
	:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 1) = 2 \cdot \frac{1}{4} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/4}{1/2 \cdot 1/4} +		:$$\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}.$$
	2 \cdot \frac{1}{4} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/4}{1/2 \cdot 1/2}
	$$
	:$$\Rightarrow \hspace{0.3cm} I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 1)
	\hspace{0.15cm} \underline {=0.5\,{\rm (bit)}}
	\hspace{0.05cm}.$$

	*The first term summarises the two horizontally shaded fields in the graph, the second term the vertically shaded fields.		*The first term summarises the two horizontally shaded fields in the graph, the second term the vertically shaded fields.
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	*Consequently, the same conditional mutual information is obtained:		*Consequently, the same conditional mutual information is obtained:
	:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 2) = I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 1)		:$$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}.$$
	\hspace{0.15cm} \underline {=0.5\,{\rm (bit)}}
	\hspace{0.05cm}.$$


	'''(3)'''  The equation is for  $Z = \{1,\ 2\}$  with  ${\rm Pr}(Z = 1) =p$  and  ${\rm Pr}(Z = 2) =1-p$:		'''(3)'''  The equation is for  $Z = \{1,\ 2\}$  with  ${\rm Pr}(Z = 1) =p$  and  ${\rm Pr}(Z = 2) =1-p$:
	:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z) = p \cdot I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 1) + (1-p) \cdot I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 2)\hspace{0.15cm} \underline {=0.5\,{\rm (bit)}}		:$$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}.$$
	\hspace{0.05cm}.$$
	*It is considered that according to subtasks  '''(1)'''  and  '''(2)'''  the conditional mutual information for given  $Z = 1$  and given  $Z = 2$  are equal.		*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  $p$.		*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  $p$.
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	*For  $Pr(Z = 1) = Pr(Z = 2) = 1/2$  the scheme sketched on the right results.		*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:		*Again, only the two horizontally shaded fields contribute to the mutual information:
	:$$ I(X;W) = 2 \cdot \frac{1}{8} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1/8}{1/2 \cdot 1/8}		:$$ 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}.$$
	\hspace{0.15cm} \underline {=0.25\,{\rm (bit)}} \hspace{0.35cm} < \hspace{0.35cm} I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z)
	\hspace{0.05cm}.$$

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

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	[[Category:Information Theory: Exercises\|^3.2 Entropies of 2D Random Variables^]]		[[Category:Information Theory: Exercises\|^3.2 Entropies of 2D Random Variables^]]
			[[de:Aufgaben:3.08_Bedingte_Transinformation]]

Noah at 08:16, 24 September 2021

2021-09-24T08:16:25Z

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	{{quiz-Header\|Buchseite=~~Informationstheorie~~/~~Verschiedene Entropien zweidimensionaler Zufallsgrößen~~		{{quiz-Header\|Buchseite=Information_Theory/Different_Entropy_Measures_of_Two-Dimensional_Random_Variables
	}}		}}

Javier: Text replacement - "generalis" to "generaliz"

2021-09-22T11:48:09Z

Text replacement - "generalis" to "generaliz"

← Older revision		Revision as of 13:48, 22 September 2021
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	The result  $I(X; W\|Z) > I(X; W)$  is true for this example, but also for many other applications:		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..		*If I know  $Z$, I know more about the 2D random variable  $XW$  than without this knowledge..
	*However, one must not ~~generalise~~ this result:		*However, one must not generalize this result:
	:Sometimes  $I(X; W) > I(X; W\|Z)$, actually applies, as in  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgr%C3%B6%C3%9Fen#Conditional_mutual_information\|Example 4]]  in the theory section.		:Sometimes  $I(X; W) > I(X; W\|Z)$, actually applies, as in  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgr%C3%B6%C3%9Fen#Conditional_mutual_information\|Example 4]]  in the theory section.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 14:31, 21 September 2021

2021-09-21T14:31:50Z

← Older revision		Revision as of 16:31, 21 September 2021
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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	[[File:P_ID2814__Inf_A_3_8a.png\|right\|frame\|2D probability functions for  $Z = 1$]]		[[File:P_ID2814__Inf_A_3_8a.png\|right\|frame\|Two-dimensional probability mass functions for  $Z = 1$]]
	'''(1)'''  The upper graph is valid for  $Z = 1$   ⇒   $W = X + Y$.		'''(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).		*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).
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	*The first term summarises the two horizontally shaded fields in the graph, the second term the vertically shaded fields.		*The first term summarises the two horizontally shaded fields in the graph, the second term the vertically shaded fields.
	*The ~~latter~~ do not contribute because of  $\log_2 (1) = 0$ .		*The second term do not contribute because of  $\log_2 (1) = 0$ .



	[[File:P_ID2815__Inf_A_3_8b.png\|right\|frame\|2D probability functions for  $Z = 2$]]		[[File:P_ID2815__Inf_A_3_8b.png\|right\|frame\|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)'''~~ ~~.		'''(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:		*Consequently, the same conditional mutual information is obtained:
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	\hspace{0.15cm} \underline {=0.5\,{\rm (bit)}}		\hspace{0.15cm} \underline {=0.5\,{\rm (bit)}}
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$
	~~<br clear=all>~~

	'''(3)'''  The equation is for  $Z = \{1,\ 2\}$  with  ${\rm Pr}(Z = 1) =p$  and  ${\rm Pr}(Z = 2) =1-p$:		'''(3)'''  The equation is for  $Z = \{1,\ 2\}$  with  ${\rm Pr}(Z = 1) =p$  and  ${\rm Pr}(Z = 2) =1-p$:
	:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z) = p \cdot I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 1) + (1-p) \cdot I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = 2)\hspace{0.15cm} \underline {=0.5\,{\rm (bit)}}		:$$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)}}
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	*If I know  $Z$, I know more about the 2D random variable  $XW$  than without this knowledge..		*If I know  $Z$, I know more about the 2D random variable  $XW$  than without this knowledge..
	*However, one must not generalise this result:		*However, one must not generalise this result:
	:Sometimes  $I(X; W) > I(X; W\|Z)$, actually applies, as in  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgr%C3%B6%C3%9Fen#Conditional_mutual_information\|~~example 3~~]] in the theory section.		:Sometimes  $I(X; W) > I(X; W\|Z)$, actually applies, as in  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgr%C3%B6%C3%9Fen#Conditional_mutual_information\|Example 4]]  in the theory section.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 14:24, 21 September 2021

2021-09-21T14:24:38Z

← Older revision		Revision as of 16:24, 21 September 2021
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	Three different definitions of mutual information are used in this exercise:		Three different definitions of mutual information are used in this exercise:
	*the ''conventional''  mutual information zwischen  $X$  and  $W$:		*the  <u>conventional</u>  mutual information zwischen  $X$  and  $W$:
	:$$I(X;W) = H(X) - H(X\|\hspace{0.05cm}W) \hspace{0.05cm},$$		:$$I(X;W) = H(X) - H(X\|\hspace{0.05cm}W) \hspace{0.05cm},$$
	* the ''conditional''  mutual information between  $X$  and  $W$  with a ''given fixed value''  $Z = z$:		* the  <u>conditional</u>  mutual information between  $X$  and  $W$  with a  <u>given fixed value</u>  $Z = z$:
	:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = z) = H(X\hspace{0.05cm}\|\hspace{0.05cm} Z = z) - H(X\|\hspace{0.05cm}W ,\hspace{0.05cm} Z = z) \hspace{0.05cm},$$		:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z = z) = H(X\hspace{0.05cm}\|\hspace{0.05cm} Z = z) - H(X\|\hspace{0.05cm}W ,\hspace{0.05cm} Z = z) \hspace{0.05cm},$$
	* the ''conditional''  mutual information between  $X$  and  $W$  for a ''given random ~~value''~~  $Z$:		* the  <u>conditional</u>  mutual information between  $X$  and  $W$  for a  <u>given random variable</u>  $Z$:
	:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z ) = H(X\hspace{0.05cm}\|\hspace{0.05cm} Z ) - H(X\|\hspace{0.05cm}W \hspace{0.05cm} Z ) \hspace{0.05cm}.$$		:$$I(X;W \hspace{0.05cm}\|\hspace{0.05cm} Z ) = H(X\hspace{0.05cm}\|\hspace{0.05cm} Z ) - H(X\|\hspace{0.05cm}W \hspace{0.05cm} Z ) \hspace{0.05cm}.$$

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	Hints:		Hints:
	*~~How large is~~ the ~~mutual information between~~  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen\|Different entropies of two-dimensional random variables]].		*The exercise belongs to the chapter  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen\|Different entropies of two-dimensional random variables]].
	*In particular, reference is made to the page  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen#Conditional_mutual_information\|Conditional mutual information]].		*In particular, reference is made to the page  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen#Conditional_mutual_information\|Conditional mutual information]].

Noah at 10:04, 14 September 2021

2021-09-14T10:04:03Z

← Older revision		Revision as of 12:04, 14 September 2021
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	From  $X$,  $Y$  and  $Z$  we form the new random variable  $W = (X+Y) \cdot Z$.		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  $I(X; W) ≠ 0$.		* 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 ~~task~~.		*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 ~~task~~:		Three different definitions of mutual information are used in this exercise:
	*the ''conventional''  mutual information zwischen  $X$  and  $W$:		*the ''conventional''  mutual information zwischen  $X$  and  $W$:
	:$$I(X;W) = H(X) - H(X\|\hspace{0.05cm}W) \hspace{0.05cm},$$		:$$I(X;W) = H(X) - H(X\|\hspace{0.05cm}W) \hspace{0.05cm},$$

Noah at 11:24, 13 September 2021

2021-09-13T11:24:55Z

← Older revision		Revision as of 13:24, 13 September 2021
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	Hints:		Hints:
	*How large is the mutual information between  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen\|Different entropies of two-dimensional random variables]].		*How large is the mutual information between  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen\|Different entropies of two-dimensional random variables]].
	*In particular, reference is made to the page  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen#~~Bedingte_Transinformation~~\|Conditional mutual information]].		*In particular, reference is made to the page  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen#Conditional_mutual_information\|Conditional mutual information]].


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	<quiz display=simple>		<quiz display=simple>

	{How large is the ~~transinformation~~ between  $X$  and  $W$,  if  $Z = 1$  always holds?		{How large is the mutual information between  $X$  and  $W$,  if  $Z = 1$  always holds?
	\|type="{}"}		\|type="{}"}
	$ I(X; W \| Z = 1) \ = \ $ { 0.5 3% } $\ \rm bit$		$ I(X; W \| Z = 1) \ = \ $ { 0.5 3% } $\ \rm bit$

	{How large is the ~~transinformation~~ between  $X$  and  $W$,  if  $Z = 2$  always holds?		{How large is the mutual information between  $X$  and  $W$,  if  $Z = 2$  always holds?
	\|type="{}"}		\|type="{}"}
	$ I(X; W \| Z = 2) \ = \ $ { 0.5 3% } $\ \rm bit$		$ I(X; W \| Z = 2) \ = \ $ { 0.5 3% } $\ \rm bit$
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	*If I know  $Z$, I know more about the 2D random variable  $XW$  than without this knowledge..		*If I know  $Z$, I know more about the 2D random variable  $XW$  than without this knowledge..
	*However, one must not generalise this result:		*However, one must not generalise this result:
	:Sometimes  $I(X; W) > I(X; W\|Z)$, actually applies, as in  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgr%C3%B6%C3%9Fen#~~Bedingte_Transinformation~~\|example 3]] in the theory section.		:Sometimes  $I(X; W) > I(X; W\|Z)$, actually applies, as in  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgr%C3%B6%C3%9Fen#Conditional_mutual_information\|example 3]] in the theory section.
	{{ML-Fuß}}		{{ML-Fuß}}

Noah at 11:23, 13 September 2021

2021-09-13T11:23:05Z

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