How is mutual information defined for two random variables X and Y?
4 Mutual information. Definition The mutual information between two discreet random variables X, Y jointly distributed. according to p(x, y) is given by. I(X;Y ) = ∑ x,y.
When X and Y are statistically independent then mutual information is?
An important theorem from information theory says that the mutual informa- tion between two variables is 0 if and only if the two variables are statistically independent. P(x, y) log P(x, y) P(x)P(y) .
When X and Y are statistically independent then the average mutual information I X Y is?
The average mutual information I(X; Y) is a measure of the amount of “information” that the random variables X and Y provide about one another. Notice from Definition that when X and Y are statistically independent, we have I(X; Y) = 0, which means that X and Y do not provide any information about one another.
How to calculate mutual information between two variables?
Definition The mutual information between two discreet random variables X,Y jointly distributed according to p(x,y) is given by I(X;Y) = X x,y p(x,y)log p(x,y) p(x)p(y) (24) = H(X)−H(X|Y) = H(Y)−H(Y|X) = H(X)+H(Y)−H(X,Y). (25) 4
What is the covariance between X and Y?
Here, we’ll begin our attempt to quantify the dependence between two random variables X and Y by investigating what is called the covariance between the two random variables. We’ll jump right in with a formal definition of the covariance.
Which is a measure of conditional mutual information?
The conditional mutual information is a measure of how much uncertainty is shared byXandY, but not byZ. 4.3 Properties
How to define the entropy of a random variable?
= H(g(X))+H(X|g(X)), (13) so we have H(X)−H(g(X) = H(X|g(X)) ≥ 0. (14) with equality if and only if we can deterministically guess X given g(X), which is only the case if g is invertible. 3 Continuous random variables. Similarly to the discrete case we can define entropic quantities for continuous random variables.