This tag is for [Entropy](http://en.wikipedia.org/wiki/Entropy_(information_theory)) in Mathematics.

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random variable and hamming distance [on hold]

Given a random variable $Z=(Z_1,Z_2,\ldots ,Z_N)$ such that the entropy of each bit is high, or overall H(Z) is close to 1. Is it it obvious to see that given another string $y$ (note it is not a ...
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29 views

What is the variance of self-information (or surprisal)?

The self-information of an outcome $x_i$, or surprisal, is defined as: $$ I(x_i)=-\log P(x_i), $$ where $P$ means probability. This way, the Shannon entropy can be seen as the "average" or "expected" ...
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31 views

Symmetric Decomposition of Mutual Information

Let's say I have a random variable $Z$ and a also collection of random variables $A$, $B$, $C$, $D$. The mutual information of these is $I(Z; A, B, C, D) = X$. Is there a good way to "decompose" $X$ ...
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1answer
14 views

Entropy of $Y=bX$

If I have two random variables $Y$ and $U$ related as $Y=bU$, where $b>0$ is a constant and knowing that $\text{H}(x)$ represents the shannon entropy, such that: $$ \text{H}(x)=−\int \text{p}(x) ...
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Why is ln(p_i) not rounded down in theexpression for Shannon entropy?

Entropy supposedly " is the average amount of information contained in each message received"(Wikipedia: Entropy). However, to calculate the Shannon entropy for a finite sample, we have the sum over ...
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1answer
25 views

data processing inequality using non-deterministic functions

Generally data processing inequality says that the entropy cannot increase on applying a function f, or to be precise $H(f(X))\leq H(X)$ (also it is reversed if we know the function is k-to-1 so there ...
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1answer
26 views

Entropy of a uniform distribution

The entropy of a uniform distribution is $ ln(b-a)$. With $a=0$ and $b=1$ this reduces to zero. How come there is no uncertainty?
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15 views

Change of bases for entropy

From Cover and Thomas, Elements of Information Theory: Why isn't it: $ \log_b(p) = \frac{\log_a(p)}{\log_a(b)} $, so that $ H_a(X) $ is multiplied with $ \frac{1}{\log_b a} $?
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Finding the Legendre transform of an “entropy type” functional

I want to find the Legendre Transform of $$T(f) = \int_{\mathbb{R}^2} f \log \left(\frac{f}g{}\right) \, dx$$ on a set $H_M = \{ f: f \ge 0 \text{ and } \int_{\mathbb{R}^2} f =M\}$, where g is some ...
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42 views

Entropy calculation of Fibonacci distribution

For any positive integer $N$, consider the Fibonacci sequence $F_n$ of length $N$. Using $F_n$ we can define a Fibonacci discrete probability distribution as follows: $$p_N(n)=\frac{F_n}{\sum_{k=1}^N ...
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38 views

Estimate uncertainty of a function

I have a question about estimate the uncertainty of a function. I have a variable $x$. I assume that I can predict the variable $x$ is followed by a function $f(x)$ such as $$f(x)=\exp(-x^2)$$ Now, ...
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70 views

Structure of equientropic transformations

Given a probability vector $v=(v_1,\ldots,v_n)$ with $1\geq v_i\geq 0$ and $\sum_{i=1}^n v_i=1$ its entropy can be defined as: $$ H(v):=-\sum_{i=1}^nv_i\log v_i $$ I wonder what is known about ...
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13 views

A simpler way for an entropy inequality

I have to show that $\frac {1}{N}H(X_1,...,X_N)\le H(X_1)$. for a stationary stochastic process. I know that $H(X_1,...,X_N)=\Sigma _{i=1}^N H(X_i|X_1,...,X_{i-1})$. So far I have plugged that ...
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1answer
26 views

Mutual Information Entropy Inequality

I am trying to prove $H(x,y:z)>H(x:z)+H(y:z)$ and here is what I have. LHS: $=H(xy)-H(xy|z)=-\Sigma p(xy)lg(p(xy))+\Sigma p(xy|z)lg( \frac{p(xyz)}{p(z)})$ RHS: ...
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20 views

Does anybody know of the proof of conditional entropy in general

I was wondering if someone could provide a proof of conditional entropy in general or with two and three variables or a place where I could find it. I am having trouble with some of the algebra and ...
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21 views

How to calculate the mutual information between two outputs of Rayleigh fading channels

We have the two channels: $$X_{a,i} = H_{i}s_{i} +N_{a,i} \\ X_{b,i} = H_{i}s_{i} +N_{b,i} $$ for $1 \leq i \leq n$, where $H_i$ denotes the i.i.d. channel coefficient and is a zero-mean complex ...
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1answer
13 views

Relative Entropy decomposition reference

may I ask for some reference pointers? My bad as I got a classic case of losing my reference and thus unsure what I wrote was right or wrong. I tried looking my old references and internet and didn't ...
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2answers
38 views

Proving certain aspects of Entropy

I am trying to prove three properties of entropy. $1)$ $H(X|Y,Z)\le H(X|Y)$ $2)$ $H(X|Y,Z)\le H(X,Y)$ $3)$ $H(X,Y,Z)+H(Y)\le H(X,Y)+H(Y,Z)$ I have proved the third one, but it is based on part 1. ...
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22 views

Motivation for the binary entropy function

What is the motivation for the definition of the binary entropy function $H(x) = -p\log_2(p) - (1-p)\log_2(1-p)$? I understand that we want the entropy to be zero at $p = 0$ and $p = 1$ (no ...
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Covering Number of Lipschitz Function Space

On page 12 of the slide deck here, the author gives an example where a lower bound (and upper bound, but I am particularly interested in the lower bound) on the $\epsilon$-covering number of a ...
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46 views

Limit of $\frac{1}{n}\log({n\choose np})$ without using Stirling's formula

I am trying to evaluate the following limit: $$ \forall p\in(0,1) ,\lim_{n\rightarrow \infty} \frac{1}{n}\log{n\choose \lfloor np \rfloor} =H(p),$$ where $\lfloor x\rfloor$ means the greatest integer ...
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Metric Entropy Upper Bounds

In the paper Information-Theoretic Determination of Minimax Rates of Convergence the authors present Theorem 3 as follows: If $M_2(\epsilon)$ is the $\ell_2$ packing entropy of a density class ...
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Question regarding the Entropy of a probability mass function

I assume that the entropy, $E$, of a probability mass function (pmf), $p(X)$, of a discrete random variable, $X$, is computed as: $$\begin{align}\mathbb{E}(p(X)) &= -p(X = x_1)\log[p(X = ...
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20 views

conditional entropies for identical distributions

Let me say I have two distributions $X$ and $Y$ which are identical, but they are not independent. Now if were to calculate the conditional entropies $H(X|Y)$ and $H(Y|X)$. Is calculating one joint ...
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Absolute value of difference between entropies (of two distributions)

I have the following inequality for the $L_1$ distance between two distributions $Q$, $Q^n$ on a finite set $B$: $$\|Q-Q^n|| < \frac{2|B|}{n}\leq \frac{C}{n} \leq \frac12 $$ Assuming $C\geq2|B|$, ...
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Shannon Entropy, prove $H(Wx)=H(x)+\log|\det W|$

I'm doing an essay on ICA (independent component analysis), and I could use some help. In essence, ICA is an algorithm that minimizes the entropy of $n$ $1$-dimensional random variables, but to show ...
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35 views

Do Redundancy and Shannon's Entropy correlate?

I wonder can anyone answer whether the Entropy and the Redundancy correlate. I am interested in cases where the distribution is over discrete categories (i.e., cases), but I think it should apply for ...
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Entropy proerty

Let $a,b,c>0$ be distinct postive reals. Define four different probability distributions: $$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ ...
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equation for entropy of angles around a circle?

Hope you can help.. I'm using an random (even) distribution of angles to represent agents converging on a target value. (e.g zero degrees, where agents are at 4, 58, 99 & 202 degrees) I need to ...
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1answer
37 views

Understanding an application of Entropy

I'm struggling with the following exercise on entropy. Suppose that your friend Alice chooses a number $X$ uniformly at random from $[1,n]$, which she writes down using $\log n$ bits; you can assume ...
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1answer
24 views

Inverse of shannon entropy

The shannon entropy of a bit $(p,1-p)$ is $$H(p)=-p\log(p)-(1-p)\log(1-p)$$. This is a well behaved function that uniquley assigns each state (up to permutation of its elements, i.e. ...
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1answer
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Entropy of floating number array

I am familiar with shanon's definition of entropy. $$ H(P) = - \sum_{i=1}^n p_i \cdot \log_2(\mathcal p_i) $$ I am today in the situation that I'd like to compute an entropy like function but for a ...
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1answer
30 views

Entropy Solution of $u_t+(u^2/2)_x=0$

Given the initial data $$ g(x)= \cases{ 1 & x< -1 \\ 0 & -1 < x< 0 \\ 2 & 0 < x< 1 \\ 0 & 1 < x \\ } $$ What is the entropy solution of $u_t+(u^2/2)_x=0$?
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1answer
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Reference for entropy of the binomial distribution?

The Wikipedia page Binomial distribution says that the entropy of the Binomial(n,p) is $\frac{1}{2}\log_2\left(2\pi e n p (1-p)\right) + O\left(\frac{1}{n}\right)$. What is a reference (paper or ...
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Entropy of beta-expansion

We have the transformation $T: [0,1) \rightarrow [0,1)$ given by $Tx = \beta x \text{ mod } 1$ with $\beta = \frac{1+ \sqrt{5}}{2}$. Calculate the entropy $h_{\mu}(T)$ of $T$ wrt the invariant ...
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Shannon Entropy Continuity Constraint

I have the following problem: I want to find the probability density $p$ which maximizes the Shannon entropy \begin{equation} S := - \int_{x_b}^{x_c} dx ~ p(x) \log (p(x)) \end{equation} under the ...
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Shannon entropy on the simplex

The state of a trit $$\{p_1,p_2,p_3=1-p_1-p_2\}$$ can be represented as a triangular simplex. The centre of the simplex is the maximally mixed state $$m=\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$$. And ...
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1answer
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Greenberg-Hastings-Model: What kind of shift space is it?

I would like to read something about the entropy of the one-dimensional Greenberg-Hastings-Model - and I think maybe I can find something about that in the book "Symbolic Dynamics and Coding" - but I ...
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1answer
28 views

Is this formula a KL divergence?

As everyone knows KL divergence's formula is $KL(p||q) = \sum_{i=1}^{n}p(i)\log (p(i)/q(i))$. In the image, formula(9) is really calculate KL(X||($(UZ^TA^T)$)) , however i have no idea why there is ...
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Summation methods and entropy

I am aware of the theory of divergent series, but don't know much of it. If you have a text to recommend, I'd be glad to hear it. Suppose I have an infinite-dimensional probability vector $\mathbf{p} ...
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Entropy of a Markov chain (right result?)

Consider the Markov chain with state space $E=\left\{0,1,2,3,4,5,6\right\}$ and transition matrix $$ \begin{pmatrix}1/5 & 3/5 & 0 & 0 & 1/5 & 0 & 0\\0 & 0 & ...
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Is there a symmetric alternative to Kullback-Leibler divergence?

I have two samples of probability distributions that I would like to compare. I have previously heard about the Kullback-Leibler divergence, but reading up on this it seems like its non-symmetricity ...
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Given a Markov chain $X \rightarrow Y \rightarrow Z$, why is $I(X;Y|Z) \leq I(X;Y)$?

A Markov chain $X \rightarrow Y \rightarrow Z$ is given, where $X,Y,Z$ are random variables characterized by the probability distribution $p(x,y,z) = p(x)p(y|x)p(z|y)$. It follows that $I(X;Y) \geq ...
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1answer
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What is the exact meaning of $I(X;Y|Z)$ in Information Theory?

I am wondering: is the notation $I(X;Y|Z)$ used to denote the mutual information between probabilities of $X$ and $Y|Z$ or between $X|Z$ and $Y|Z$?
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1answer
38 views

Entropy determination regarding lossless data compression

Suppose I had many computer data files to compress losslessly and wanted to know what is the theoretical limit to each one as far as minimum filesize possible. How would a math person go about doing ...
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Entropy of Sum vs Difference of Random Variable

I am looking for a proof of the following fact Let X and X' be i.i.d on {0,1,2}(not necessarily uniform). Prove that $H(X + X' mod\;3) \leq H(X - X' mod\;3)$ where $H()$ is the standard Shannon ...
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If $g$ is a function of the random variable $X$, is it true that $H(X) = H(X) + H(g(X)\mid X)$?

I think my homework about entropy is formulated incorrectly. The question is the following: let $X$ be a discrete random variable. Show that the entropy of a function $g$ of $X$ is less than or ...
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Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...
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Entropy of convolution of measures

Let $G$ be a countable, discrete group, and let $\mu_1,\mu_2$ be probability measures on the group $G$. We define the entropy of $\mu_i$ as $H(\mu_i)=\sum\limits_{g \in G}-\mu_i(g)\log(\mu_i(g))$ ...
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1answer
38 views

Renyi entropy (zeroth order)

I am reading a book on information theory, therein has been introduced Renyi entropy of order $\alpha$ as $S_{\alpha} = \frac{1}{1-\alpha}\log(Tr\rho^{\alpha})$, where $\rho$ is density matrix. It ...