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

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142 views

Approximation of Shannon entropy by trigonometric functions

Define Shannon entropy by $$I(p) = -p \log_2 p$$ Numerical experimentation shows that $\sin(\pi p)^{1-1/e}$ is a good approximation to $I(p) + I(1-p)$ on $[0,1],$ never differing by more than 3.3%. ...
5
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98 views

Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
5
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81 views

Decrease of entropy when iterating a random discrete function

Let $m$ be a positive integer. Let $S$ be the set of non-negative integers $x$ less than $m$, with $|S|=m$. Let $X_0$ be the discrete uniform distribution over $S$, with $P(x)=\begin{cases} 1/m & ...
4
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27 views

Arnold's combinatorial description of entropy.

V.I. Arnold says that entropy is related to the asymptotic behaviour of polynomial coefficients. This is mentioned in his book "Dynamics, Statistics and Projective Geometry of Galois Fields". Here ...
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93 views

A combinatorial problem.

Let be $(X, \mathbb{A}, \mu)$ a measure space, a partition of $ X $ is a disjoint family $\xi=\{P_1,\ldots,P_k \}$ of measurable sets such tath $\bigcup P_i=X\pmod0).$ If $\xi=\{P_1,\ldots,P_k ...
3
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49 views

Computation of a sum using Stirling's approximation and Watson's lemma

$$Ω=\sum_{n=0}^{N-\frac{E}{\epsilon}} \frac{Ν!}{\left(\frac{N-n-\frac{E}{\epsilon}}{2}\right)!\left(\frac{N-n+\frac{E}{\epsilon}}{2}\right)!n!}$$ I am supposed to calculate the above sum using first ...
3
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109 views

Very special geometric shape (No name yet?)

I suppose this geometric shape is something very 'special'. I cannot clarify in short about being 'special', but I think this shape stands together with such special shapes like the square and the ...
3
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111 views

Huffman codes: does less entropy imply less weighted average codeword length?

Let $\Sigma$ be a source alphabet with a probability distribution over its symbols $P$. Then, the Shannon entropy of $\Sigma$ is $$-\sum p_j \times -\mbox{log}_2(p_j)$$ where $p_j$ is the probability ...
3
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103 views

Topological applications of topological entropy

I just learned topological entropy during a lecture about dynamical systems, and I wonder whether there exist purely topological applications of it.
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119 views

Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it ...
3
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109 views

Reference for a transformation

Has the (Lebesgue-)ergodic transformation $T: \{0,1\}^{\mathbb{N}} \to \{0,1\}^{\mathbb{N}}$ defined by $T(x(0)x(1)x(2)\cdots) = x(1)x(3)x(5)\cdots$ been well-studied? If so, where? Does it have a ...
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96 views

Which takes more energy: Shuffling a sorted deck or sorting a shuffled one?

You have an array of length $n$ containing $n$ distinct elements. You have access to a comparator on the elements (a black-box function that takes $a$ and $b$ and returns true if $a < b$, false ...
3
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462 views

Can I map entropy values to a common range so that they are comparable?

I am using the standard Shannon entropy formula for calculating the entropy of a system at different states. The system has a different number of possible outcomes at each state, in other words the ...
2
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25 views

Differential entropy of the product of Gaussian random variables

Given two independent Gaussian random variables $X \sim \mathcal{N}(\mu_x,\sigma_x^2)$ and $Y \sim \mathcal{N}(\mu_y,\sigma_y^2)$. We look at the product distribution of these two random variables ...
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30 views

How to evaluate the quality of the probability distribution output of a classifier?

In a classification problem, I have trained a neural network which outputs class probabilities for a given input. For a new input, I now want to evaluate the "quality" of the neural network's ...
2
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38 views

Is the topological entropy of a continuous map $T\colon X\to X$ zero if $X$ is a finite topological space?

Let $X$ be a finite topological space and $T\colon X\to X$ continuous. As the title already suggests, I am wondering if the topological entropy of $T$, denoted by $h(X,T)$, then is $0$. As far as I ...
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20 views

Graph Entropy - A Tractable Measure to Measure Distinguishability of Neighbourhoods

Given a labelled directed graph G, I am interested in a measurement of G that captures how distinguishable arbitrary connected sub graphs of G are. Labels may repeat and as such two or more different ...
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44 views

Topological Entropy of $T$ on subset $Y\subset X$

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the following dynamics described by $T\colon X\to X$ as follows: A 1 becomes a 2, a 2 becomes a 0 and a 0 becomes a 1 if at least one of its two ...
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87 views

Is it true that $\sup_{y'\in Y'}h_d(T,U^{-1}(y'))=0$, i.e. $h(T')=h(T)$? (Bowen, Topological entropy)

First I have to give the background to my question: Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the map $T\colon X\to X$ which describes the following dynamics: For $x\in X$, which is a ...
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81 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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34 views

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 ...
2
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66 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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87 views

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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38 views

intuition for entropy solutions

For a hyperbolic PDE of the form $$u_t + f(u)_x = 0$$ it turns out that the right notion of solution is entropy solution. Now, the notion of classical solutions are obviously very natural, and also ...
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71 views

Maximize the expected Brier Score of P relative to P.

Fix a finite set $\mathcal S$ and let $\mathcal P$ be the collection of probability functions over the Boolean closure of $\mathcal S$. Let $\beta : \mathcal P \times \mathcal S \to \mathbb R$, with ...
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54 views

Joint entropy maximization with a constraint

So I have 2 random variables X and Y, where X can take on values {0,1,2,3} and Y can take on values {0,1,2,3,4}. I need to maximize H(X,Y) subject to the constraint that P(X≠Y)=0.5. This also gives ...
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72 views

Entropy of sum of random variables

Let $x_1,x_2,\dots,x_n$ by random variables which take the values $0$ or $1$ with $P(x_i = 1) = p_i$ and $P(x_i = 0) = 1-p_i$, where $0 \leq p_i \leq 1$ for $i=1,2,\dots, n$. Let $$X= \sum_{i=1}^n ...
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27 views

probability subspaces that make entropy function equal to a constant value

Given the entropy fucntion: $$ H = -\sum_i^n p(i) \ln(p(i))\,.$$ where $p(i)$ are probabilities and $n=4$, I would like to know all the points in the probability space that make $H = k$, being $k$ a ...
2
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205 views

How many code words if average code length equals entropy

I've been given a proof of the following: If $q\geq2$, then there is a source $S$ with $q$ symbols, and an instantaneous $r$-ary code $C$ satisfying $L(C)=H_r(S)$ if and only if $q\equiv 1 ...
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77 views

Optimization problem in the Von Neumann Entropy

I have a constrainted optimization problem in the Von Neumann Entropy. In a CVX-like syntax the problem goes as follows: given variable $\mathtt{c(n)}$ $$\begin{align} \text{minimize} \qquad & ...
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67 views

Entropy of a measure-preserving transformation

Let $T:X \rightarrow X$ be a measure-preserving transformation of the probability space $(X,B,\mu)$. ($X$ is a topological space). If $K\subset X$ is a compact and invariant set, show that ...
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70 views

Computing Relative entropy?

I am doing a project for my CS class and I was wondering if the following would work. I have 50 different people who have rated the same 50 books. The rating system is as follows: negative 5 = hate ...
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176 views

rényi entropy as a derivative

Let $x=(x_i)$ be a probability measure on $\{1,\ldots,n\}$. Suppose $1<p<\infty$. The Rényi entropy of $x$ is $$ H^p(x)=\frac{1}{1-p}\log \sum_{i} x_i^p. $$ Does there exist a formula for ...
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39 views

Entropy and the probability to guess

Let $X$ be a discrete random variable and suppose that we choose a random value $X=x_1$. Let $A$ be an event such that $H[ X \mid A] = k$, where $$H[X \mid A] = - \sum_{x} P[X =x \mid A] \log_2( P[X ...
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30 views

How to show the inf can be achieved by some nonnegative $u\in H^1(M)$?

When I read some about Perelman's $\mathcal W$ function, I get stuck with the red line in the picture below.Seemly, I should to read the 8.2 Existence of minimizers of Evans' PDE. But I am not sure , ...
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18 views

Show entropy bound unit simplex

Let $\mathcal{S}_d$ be the $d$-dimensional unit simplex. Then for the norm $||x||_1 = \sum_i |x_i|$ and $0 < \varepsilon \leq 1$, $$N(\varepsilon, \mathcal{S}_d, ||\cdot||_1) \leq ...
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15 views

Efficient numerical calculation of Entropy of Poisson RV

There doesn't exist a closed-form solution of following infinite series (entropy of Poisson RV): $$ \displaystyle \sum_{x=0}^{\infty} \Big( \frac{e^\lambda \lambda^x}{x!} \Big) \cdot \text{log}_2\Big( ...
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51 views

Mutual information of independent fair binary random variables

Let random variables $X,Y$ independent fair random variables that take the values 0 and 1 with equal probability and $Z=X+Y$. So, $I(X;Y)=0$ and I am trying to find their conditional mutual ...
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35 views

Asymptotic binomial ratios

I am in need of asymptotic version of $$\frac{ \displaystyle \binom{n^{1-s}}{n^s}}{\displaystyle \binom{n}{n^{s}}}$$ where $n\in\Bbb N$ and $s\in\big(0,\frac12\big)$ and $$\displaystyle \frac{ ...
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28 views

Mutual information between 2 sequences of random variables?

How would I go about expanding $I(X_1,...,X_n;Y_1,...,Y_n)$? The chain rule exists for a single case, i.e.: $I(X_1,...,X_n;Y)=\sum^n_{i=1} I(X_i;Y|X_{i-1},...,X_1)$, but I'm having doubts if this can ...
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21 views

Source coding with 2 distinct distributions and entropies

I'm learning about source coding, and many of the books/resources I've read give examples of the source $X^n$ being defined as a sequence of iid random variables. How about when the sequence is ...
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13 views

Are there any other measures of complexity for a continuous map than topological entropy?

Let $X$ be a compact topological space and $T\colon X\to X$ be continuous. In order to say something about the complexity of $(X,T)$ there is of course the notion of topological entropy of $T$, ...
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17 views

Entropy of a 2D function versus 1D function.

I am a novice in information theory so this is more of a question seeking pointers to ideas/references to think further on the thought. I want to make concrete the idea that a function of two ...
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28 views

How to choose assymetry for KL divergence?

I have two 2D probability distributions of eye movements of two different images. Suppose I call the first distribution of Image 1: $P$, and the second distribution of image 2: $Q$. Since ...
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43 views

Intuition for Entropy over Fractals

Is there intuition for "mathematical" entropy. I know that physical entropy tracks the order in a dynamical system, for thermodynamics. As entropy goes up, general randomness and disorder goes up. ...
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13 views

Integrating entropy on an arbitrary boundary

Entropy, denoted as H, is: $$ H = -\int_a^b p\ln(p) dx $$ where the range a to b is some arbitrary boundary and where p is given by the classic: $$ p(x) = ...
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48 views

Cross-entropy minimization - equivalent unconstrained optimization problem

I'm looking at this paper "An Alternative Method for Estimating and Simulating Maximum Entropy Densities" ...
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38 views

Proof of a fact about mutual information and entropy

It is needed to prove that for distribution (a, b, c) such that $I(a;b|c) = I(a;c|b) = I(b;c|a) = 0$ exists a random variable $d$ such that $H(d) = I(a, b, c)$ and $a,b,c$ are independent with ...
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21 views

Maximum entropy distribution given second order marginals

Let $p(x,y,z)$ be a probability distribution over 3 variables (suppose them discrete, but it shouldn't matter). I know that the distribution with maximal entropy which preserves the first order ...
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87 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 ...