# What is chain rule of entropy?

## What is chain rule of entropy?

Abstract. A chain rule for an entropy notion H(·) states that the en- tropy H(X) of a variable X decreases by at most l if conditioned on an l-bit string A, i.e., H(X|A) ≥ H(X) − l. More generally, it satisfies a chain rule for conditional entropy if H(X|Y,A) ≥ H(X|Y ) − l.

Is entropy always nonnegative?

The relative entropy is always non-negative and zero if and only if p = q. Note that the relative entropy is not a true metric, since it is not symmetric and does not satisfy the triangle inequality. The function is said to be strictly convex if equality holds only if λ = 0 or λ = 1.

Is entropy always less than 1?

Entropy is measured between 0 and 1. (Depending on the number of classes in your dataset, entropy can be greater than 1 but it means the same thing , a very high level of disorder.

### How do you find the sum of entropy?

The sum of the entropies of 2 independent random variables is the entropy of their joint distribution, i.e. H(X,Y)=H(X)+H(Y) . This implies that in this particular case H(X,Y)=(ln(2πeσ2)/2)⋅2.

Why is entropy concave?

Because conditioning reduces the uncertainty, H(Z) ≥ H(Z|b). This proves that the entropy is concave. Also, X → Y → Z ⇐⇒ Z → Y → X. Now let us consider the property of Markov chain.

Is entropy discrete or continuous?

The actual continuous version of discrete entropy is the limiting density of discrete points (LDDP). Differential entropy (described here) is commonly encountered in the literature, but it is a limiting case of the LDDP, and one that loses its fundamental association with discrete entropy.

#### Why log is used in entropy?

Why? Because if all events happen with probability p, it means that there are 1/p events. To tell which event have happened, we need to use log(1/p) bits (each bit doubles the number of events we can tell apart).

What is the relation between joint and conditional entropy?

The joint entropy represents the amount of information needed on average to specify the value of two discrete random variables. is the conditional entropy of Y given X. The conditional entropy indicates how much extra information you still need to supply on average to communicate Y given that the other party knows X.

How do you prove entropy is concave?

4.4 Is entropy concave? In order to prove whether entropy is concave or not, we need to show following: H(λp + (1 − λ)q) ≥ λH(p) + (1 − λ)H(q) (1) 3-3 Page 4 Proof Let us assume that x ∼ p and y ∼ q on set Ω. Also, let us define another variable b with following distribution.

## Is entropy function convex?

For example, the negative Shannon entropy is a strictly convex functional and minimization of the negative Shannon entropy under linear constraints gives one of the results of the maximum entropy method .

Can entropy of a distribution be negative?

Robert B. 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 −∞.

Can differential entropy be negative?

Variants. As described above, differential entropy does not share all properties of discrete entropy. For example, the differential entropy can be negative; also it is not invariant under continuous coordinate transformations.

### Is entropy always positive?

We know that the entropy is zero for reversible processes and always positive for irreversible processes.

How entropy is measured?

The entropy of a substance can be obtained by measuring the heat required to raise the temperature a given amount, using a reversible process. The standard molar entropy, S°, is the entropy of 1 mole of a substance in its standard state, at 1 atm of pressure.

Is entropy just probability?

In classical thermodynamics, entropy is defined in terms of macroscopic measurements and makes no reference to any probability distribution, which is central to the definition of information entropy.

#### What is the difference between entropy and conditional entropy?

Entropy measures the amount of information in a random variable or the length of the message required to transmit the outcome; 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.

What is the chain rule for entropy of two experiments?

H(X;Y) =H(X)+H(YjX) =H(Y)+H(XjY) \\entropy of two experiments” Dr. Yao Xie, ECE587, Information Theory, Duke University 2 Chain rule for entropy Entropy for a collection of RV’s is the sum of the conditional entropies More generally:H(X1;X2; ;Xn) = ∑n i=1H(XijXi1; ;X1) Proof:

What is the formula to prove the relative entropy theorem?

Very handy in proof: e.g., proveD(pjjq): D(pjjq) = ∑ p(x)log p(x) q(x) x p(x))log ∑ xp(x) ∑ xq(x) = 1log1 = 0: Dr. Yao Xie, ECE587, Information Theory, Duke University 23 Convexity of relative entropy Theorem.

## What is the entropy for a collection of RVs?

Entropy for a collection of RV’s is the sum of the conditional entropies More generally:H(X1;X2; ;Xn) = ∑n i=1H(XijXi1; ;X1) Proof:

What is the formula for entropy of two experiments?

H(X;Y) =H(X)+H(YjX) =H(Y)+H(XjY) \\entropy of two experiments” Dr. Yao Xie, ECE587, Information Theory, Duke University 2

• October 23, 2022