\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
deepdream of
          a sidewalk
Show Question
Joint entropy of two random variables

For two ensembles, \( X = (x, A_x, P_x) \) and \( Y = (y, A_y, P_y) \), where there may be dependency between \(P_x \) and \(P_y \), the joint entropy of \(X\), \(Y\) is:

\[H(X, Y) = \sum_{x \in A_x} \sum_{y \in A_y} P(x, y)log \frac{1}{P(x, y)} \]

Entropy is additive for independent random variables.


\[ \begin{align*} H(X, Y) &= \sum_{x \in A_x} \sum_{y \in A_y} P(x)P(y)log\frac{1}{P(x)P(y)} \\ &= \sum_{x \in A_x} \sum_{y \in A_y} P(x)P(y)log\frac{1}{P(x)} +\sum_{x \in A_x} \sum_{y \in A_y} P(x)P(y)log\frac{1}{P(y)} \\ &=\sum_{x \in A_x}P(x)log\frac{1}{P(x)} + \sum_{y \in A_y} P(y)log\frac{1}{P(y)} \text{ (the first sum's terms are independent of y, and the second's independent of x)}\\ &= H(X) + H(Y)\end{align*} \]