What is meant by conditional entropy?

What is meant by conditional entropy?

The conditional entropy measures how much entropy a random variable X has remaining if we have already learned the value of a second random variable Y. It is referred to as the entropy of X conditional on Y, and is written H(X∣Y).

What is the difference between joint entropy and conditional 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 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.

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 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).

What is the chain rule of entropy?

With equality if and only if the Xi are independent. Proof: By the chain rule of entropies: We know that the conditional entropy of a random variable X given another random variable Y is zero if and only if X is a function of Y. Hence we can estimate X from Y with zero probability of error if and only if H(X|Y) = 0.

What is joint and conditional probability?

Joint probability is the probability of two events occurring simultaneously. Marginal probability is the probability of an event irrespective of the outcome of another variable. Conditional probability is the probability of one event occurring in the presence of a second event.

Is Shannon entropy always positive?

The Shannon entropy of the exponential distribution can be negative. 1+log(μ), Ash in his 1965 paper Information Theory (page 237) noted this: unlike a discrete distribution, for a continuous distribution, the entropy can be positive or negative, in fact it may even be +∞ or −∞.

What is conditional entropy machine learning?

Conditional Entropy. Intuitively, this is the average of the entropy of Y given X over all possible values of X. Considering the fact that (X, Y) ~ p(x, y), the conditional entropy can also be expressed in terms of expected value.

Can relative entropy negative?

Relative entropy is always non-negative as we shall see below and is used to measure learning quantitatively.

Why entropy is non negative?

It is perhaps intuitive that the entropy should be non- negative because non-negativity implies that we always learn some number of bits upon learning random variable X (if we already know beforehand what the outcome of a random experiment will be, then we learn zero bits of information once we perform it).