# What is chain rule of entropy?

Table of Contents

## 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 [3].

**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