What is the mutual information and conditional entropy?

= 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. 4.3 Properties. • Chain rule: We have the following chain rule.

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.

What is the mean by information rate define conditional entropy joint entropy?

joint entropy is the amount of information in two (or more) random variables; conditional entropy is the amount of information in one random variable given we already know the other.

Is conditional entropy less than entropy?

The conditional entropy H(X|Y ) is the amount of uncertainty that Bob has about X given that he already possesses Y . Figure 1(b) depicts this interpretation. The above interpretation of the conditional entropy H(X|Y ) immediately suggests that it should be less than or equal to the entropy H(X).

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.

Why information is entropy?

Information entropy is a concept from information theory. It tells how much information there is in an event. In general, the more certain or deterministic the event is, the less information it will contain. More clearly stated, information is an increase in uncertainty or entropy.

Why does conditioning reduce entropy?

Conditioning reduces entropy H(X|Y ) ≤ H(X) with equality of and only if X and Y are independent. Knowing another random variable Y reduces (on average) the uncertainty of variable X. X → Y → Z implies that Z → Y → X.

Is conditional entropy always positive?

Properties. Unlike the classical conditional entropy, the conditional quantum entropy can be negative. This is true even though the (quantum) von Neumann entropy of single variable is never negative.

Is mutual information bounded?

The mutual information is bounded from above by the Shannon entropy of probability distributions for single parties, i.e. I(X,Y)≤min[H(X),H(Y)] .

Why is mutual information Positive?

Mutual information is nonnegative, i.e. I(X;Y ) ≥ 0. Equivalently, H(X|Y ) ≤ H(X). Hence conditioning one random variable on another can only decrease entropy. Equality holds if and only if the random variables are independent.

What is the relationship between mutual information and entropy?

Mutual Information and Entropy. It follows from definition of entropy and mutual information that I(X;Y) = H(X) H(XjY): The mutual information is the reduction of entropy of X when Y is known.

What are the different types of entropy?

There are several other concepts of entropy, for example, relative entropy, conditional entropy, and mutual information. See, for example, Cover and Thomas (1991) and Nielson and Chuang (2000). It is easy to see that 1.

What is the conditional entropy of X and Y?

x,y. p(x,y)logp(x|y) = −E[ log(p(x|y)) ] (5) The conditional entropy is a measure of how much uncertainty remains about the random variable X when we know the value of Y. 2.3 Properties. The entropic quantities defined above have the following properties: • Non negativity: H(X) ≥ 0, entropyisalwaysnon-negative.

Can we use conditional entropy as a gauge of information gain?

In particular, the conditional entropy has been successfully employed as the gauge of information gain in the areas of feature selection (Peng et al., 2005) and active recognition ( Zhou et al., 2003 ). Accordingly, we use conditional entropy to define our scheduling criterion.