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===Nonnegativity 非负性===

===Nonnegativity 非负性===

The joint entropy of a set of random variables is a nonnegative number.

The joint entropy of a set of random variables is a nonnegative number.

一组随机变量的联合熵是一个非负数。

一组随机变量的联合熵是一个非负数。

:<math>\Eta(X_1,\ldots, X_n) \geq 0</math>

 

:<math>H(X,Y) \geq 0</math>

 

:<math>H(X_1,\ldots, X_n) \geq 0</math>

 

 

=== Greater than individual entropies 大于单个熵===

=== Greater than individual entropies 大于单个熵===

The joint entropy of a set of variables is greater than or equal to the maximum of all of the individual entropies of the variables in the set.

The joint entropy of a set of variables is greater than or equal to the maximum of all of the individual entropies of the variables in the set.

一组变量的联合熵大于或等于该组变量的所有单个熵的最大值。

一组变量的联合熵大于或等于该组变量的所有单个熵的最大值。

:<math>\Eta \bigl(X_1,\ldots, X_n \bigr) \geq \max_{1 \le i \le n}  

:<math>H(X,Y) \geq \max \left[H(X),H(Y) \right]</math>

 

:<math>H \bigl(X_1,\ldots, X_n \bigr) \geq \max_{1 \le i \le n}  

     \Bigl\{ \Eta\bigl(X_i\bigr) \Bigr\}</math>

     \Bigl\{ \Eta\bigl(X_i\bigr) \Bigr\}</math>

=== Less than or equal to the sum of individual entropies 小于或等于单个熵的总和===

=== Less than or equal to the sum of individual entropies 小于或等于单个熵的总和===

The joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set.  This is an example of [[subadditivity]].  This inequality is an equality if and only if <math>X</math> and <math>Y</math> are [[statistically independent]].<ref name=cover1991 />{{rp|30}}

The joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set.  This is an example of [[subadditivity]].  This inequality is an equality if and only if <math>X</math> and <math>Y</math> are [[statistically independent]].<ref name=cover1991 />{{rp|30}}

:<math>\Eta(X,Y) \leq \Eta(X) + \Eta(Y)</math>

 

 

:<math>H(X,Y) \leq H(X) + H(Y)</math>

:<math>\Eta(X_1,\ldots, X_n) \leq \Eta(X_1) + \ldots + \Eta(X_n)</math>

:<math>H(X_1,\ldots, X_n) \leq H(X_1) + \ldots + H(X_n)</math>

==Relations to other entropy measures==

==Relations to other entropy measures==