Is conditional mutual information symmetric?

Is conditional mutual information symmetric?

The conditional entropy H(X|Y) represents the average entropy of X when the value of Y is known. The three forms of Eq. (2) above each illustrate a particular interpretation of the mutual information. As the third form shows explicitly, mutual information is symmetric in its arguments.

Why is mutual information symmetric?

The mutual information (MI) between two random variables captures how much information entropy is obtained about one random variable by observing the other. Since that definition does not specify which is the observed random variable, we might suspect this is a symmetric quantity.

What is the relation between entropy and mutual information?

Thus, if we can show that the relative entropy is a non-negative quantity, we will have shown that the mutual information is also non-negative. = H(X|Z) − H(X|Y Z) = H(XZ) + H(Y Z) − H(XY Z) − H(Z). The conditional mutual information is a measure of how much uncertainty is shared by X and Y , but not by Z.

What is conditional entropy in information theory?

In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known. Here, information is measured in shannons, nats, or hartleys. The entropy of conditioned on is written as .

How do you find conditional entropy?

Relating mutual entropy to conditional entropy H(YX) = H(Y) iff X and Y are independent. Intuitively, uncertainty in (X,Y) is the uncertainty in X plus the uncertainty in Y given X. H(YX) = H(X,Y) − H(X) is as an alternative definition for H(YX).

Can conditional entropy negative?

Unlike the classical conditional entropy, the conditional quantum entropy can be negative. Positive conditional entropy of a state thus means the state cannot reach even the classical limit, while the negative conditional entropy provides for additional information.

Is the mutual information of a variable equal to entropy?

However mutual information of a variable with itself is equal to entropy of this variable $$ I(X, X) = H(X)$$ and is called self-information. This is true since $H(X|Y) = 0$ if values of $X$ are completely determined by $Y$ and this is true for $H(X|X)$.

How is entropy related to the base of the algorithm?

We also say that H(X) is approximately equal to how much information we learn on average from one instance of the random variable X. Note that the base of the algorithm is not important since changing the base only changes the value of the entropy by a multiplicative constant.

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.

How is the relative entropy related to the likelihood ratio?

The relative entropy is a measure of the distance between two distribu-tions. In statistics, it arises as an expected logarithm of the likelihood ratio. The relative entropy D(p||q)is a measure of the inefficiency of assuming that the distribution is q when the true distribution is p.Forexample,if