The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
0answers
62 views

How many samples to accurately estimate the entropy

Consider a discrete random variable $N$ with range $\{1,\dots,1000\}$, say. How many samples do you need to take to be sure that you can, at least in principle, compute the entropy Shannon entropy ...
1
vote
0answers
27 views

Characterization of the optimal solution in relative entropy minimization

The following optimization problem is related to relative entropy and to the limit of the iterative proportional fitting procedure. For $1 \leq i,j \leq n$ and fixed $w_{ij} \geq 0$, and fixed $a_i, ...
0
votes
1answer
59 views

Invariance of mutual information

Let $I(X,Y):=H(X)+H(Y)-H(X,Y)$ be the mutual information of the joint probability distribution $p_{XY}$ (here $H(\cdot)$ is the Shannon entropy of its argument). I know that the mutual information is ...
2
votes
0answers
107 views

On the volume entropy of negatively curved manifolds

Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the ...
0
votes
1answer
76 views

Why private randomization does not help the Shannon's source coding

I am wondering why stochastic encoder and decoder can not help the Shannon source coding? I know the achievability scheme of source coding, which is based on typicality, is deterministic, and hence we ...
9
votes
2answers
284 views

Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...
10
votes
2answers
159 views

Intrinsic significance of differential entropy

Many commentators (e.g. Jaynes, Rota) argue that the notion of "differential entropy" is problematic (as commonly defined by $ h(X) = \int ( \log\frac{1}{p(x)} ) p(x) \, dx $, where $X$ is a random ...
1
vote
3answers
115 views

How to estimate the entropy of a distribution on a power set?

Given a probability distribution $(X,p)$, its entropy is defined as $H=-\sum_{x\in X} p(x)\log p(x)$. Given a sample of observations $x_n,n=1..N$, one can estimate $p(x)=\frac{\#\{i:x_i=x\}}{N}$ and ...
1
vote
1answer
67 views

Connection between inf-entropy rate and min-entropy

I am reading the paper "Generating random bits from an arbitrary source: fundamental limits" by Vembu and Verdu. This paper is written in the language of information theory, however, I need to ...
2
votes
0answers
27 views

largest size for a randomness extractor

I am not so expert in theoretical computer science, so sorry if the question is trivial, i just could not find it in literature. Suppose we have a source $X$ with min-entropy $\ell$, the randomness ...
11
votes
0answers
149 views

Entropy in elimination theory, or a brief remark by Gelfand-Kapranov-Zelevinsky

In the introduction to their book "Discriminants, resultants and multidimensional determinant", the authors state a very intriguing observation concerning the coefficients of monomials appearing in ...
3
votes
1answer
128 views

General additive function of probability

Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that: $H$ is continuous, $H$ is symmetric w.r.t. the order of its arguments, ...
2
votes
0answers
48 views

Private Randomness extractor

Suppose we are given two random variables $X$ and $Y$ with fixed marginal and joint distribution. What is the maximum randomness that we can extract from $Y$ that is independent from $X$, that is, if ...
5
votes
0answers
174 views

Maximizing Renyi entropy for a certain channel

The channel under consideration is $T = A + B$, where $A$ and $B$ take on values in $\{0, 1\}$ according to a probability mass function. Let (joint) random vector $(A_1, A_2,\ldots, A_n)$ be denoted ...
2
votes
0answers
108 views

metric entropy for Lipschitz functions

Suppose $(X,d)$ is a metric space of unit diameer and let $F$ be the collection of all $1$-Lipschitz functions mapping $X$ to $[-1,1]$, equipped with the sup-norm $||\cdot||_\infty$. I am interested ...
8
votes
2answers
340 views

Inequality in information theory

I am reading the paper "chain independence and common information" (http://ttic.uchicago.edu/~yury/papers/independ.pdf). In this paper, an inequality is used several times (without proof) which looks ...
0
votes
1answer
151 views

Approximation of the sum involving binary entropy function

Given the following sum: $S(n) = \sum_{i=1}^{n} \frac{1}{(1-\operatorname{H}(p))^i}$ where $H$ is the binary entropy function defined as: $\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $. Let ...
3
votes
1answer
366 views

Simple reason that a mathematician cannot do better than random when guessing contents of a box?

I have a question about the finite analog of the puzzle proposed here involving mathematicians guessing the contents of boxes. Specifically, suppose there are $k$ unopened boxes each containing a ...
2
votes
1answer
108 views

Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts also have entropy s?

A subshift is a subset $X$ of $A^\mathbb{N}$ or $A^\mathbb{Z}$ (with $A$ finite), such that $X$ is topologically closed and closed under the shift operation. The shift operation is defined by ...
16
votes
1answer
672 views

Gini Coefficient and Renyi Entropy

Gini coefficient (aka Gini Index) is a quantity used in economics to describe income inequality. It is 0 for uniformly distributed income, and approaches 1 when all income is in hands of one ...
-3
votes
1answer
92 views

“logical distance” link algorithmic complexity to statistical information [closed]

Someone mentioned what I think was referred to as 'logical distance'. My hard drive crashed and I dont have the link anymore. I do recall that I ran across it on this site, in response to linking ...
2
votes
0answers
121 views

Reference for and Properties of the $alpha$-entropy

Let $T : X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost ...
2
votes
0answers
88 views

order of convergence of the conditional entropy (3)

I'm sorry for having open two questions which have been solved by elementary counter-examples provided by @AnthonyQuas. Actually I'm not an expert in information theory and I expected that a positive ...
1
vote
1answer
94 views

order of convergence of the conditional entropy (2)

Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g_n(X_n)$, then by Fano's inequality ...
3
votes
1answer
119 views

Example of Girsanov change of density with finite relative entropy, but with infinite integral over squared changed drift

Let $(\Omega, (\mathcal F_t), \mathbb P)$ denote the usual Wiener space where $\Omega = C[0,\infty)$, etc., and where $(W_t)_{t \geq 0}$ denotes the Wiener process. Let $Z \in L^1(\mathbb P)$ with $Z ...
3
votes
0answers
77 views

Is there a known generalization of the Schmidt decomposition based on a maximal set of “locally orthogonal” vectors?

I came across the following unusual generalization of the Schmidt decomposition in my work, which I describe below. I would like to know if this structure has been studied before so I can read more ...
18
votes
2answers
1k views

An Entropy Inequality (generalized)

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability ...
30
votes
1answer
2k views

An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
4
votes
3answers
437 views

Incremental entropy computation

After a quick internet search I found no method for incremental entropy computation. Question 1 Let $\{x_i\}_{i=1}^n$ and $\{x_i\}_{i=1+n}^{n+m}$ be two samples and let $S_i^j:=\sum_{k=i}^j x_k$. ...
3
votes
3answers
751 views

The relations between the Perelman's entropy functional and notions of entropy from statistical mechanics

I am looking for the relations and analogies between the Perelman's entropy functional,$\mathcal{W}(g,f,\tau)=\int_M [\tau(|\nabla f|^2+R)+f-n] (4\pi\tau)^{-\frac{n}{2}}e^{-f}dV$, and notions of ...
0
votes
1answer
101 views

order of convergence of the conditional entropy

Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g(X_n)$ then by Fano's inequality ...
9
votes
1answer
546 views

Entropy of edit distance

The edit or Levenshtein distance between two strings is the minimum number of single character insertions, deletions and substitutions to transform one string into another. If we take random binary ...
1
vote
1answer
303 views

Convexity and semicontinuity of the relative entropy function

There are several different definitions of relative entropy, and some of them are not equivalent. Following is the definition we will use in this question. Let $M$ be a closed manifold and ...
1
vote
1answer
265 views

Calculate channel capacity of general channel under constraint

Hi! Given a conditional distribution $P_{Y|X}$ I'd like to find the prior distribution $P_X$ that maximizes the mutual information $I(X;Y)$ with $P_Y(y)=\int P_{Y|X}(y|x)P_X(x)\text{dx}$ (this ...
0
votes
0answers
201 views

Calculating entropy of adjacency matrix using eigenvalue decomposition?

How to calculate entropy using the eigenvalues when the eigenvalues are negative? Is there a simple relation between the entropy of a matrix and its characteristic polynomial?
3
votes
2answers
144 views

Estimate entropy of a binary process in terms of decay of correlations

Suppose $( X_{n} )$ is an ergodic binary process with $$ \mathbb P(X_{n}=1)= \mathbb P(X_{n}=0)=\frac 12. $$ Naturally the entropy (rate) $h(X)$ of $X=(X_{n})$ satisfies $$ h(X)=\lim_{n\to\infty} ...
7
votes
1answer
804 views

What's the maximum entropy probability distribution given bounds [a,b] and mean?

What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution? For example: if a=0, b=1, ...
2
votes
0answers
109 views

A.G. Vitushkin's “Easily representable families of functions” - can it be generalized?

Background In his monograph "Estimation of the complexity of the tabulation problem" (translated into English as "Theory of the Transmission and Processing of Information") Vitushkin studies ...
2
votes
2answers
249 views

probability measures with entropy equal to nonnegative number

Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) that equals this ...
4
votes
1answer
298 views

Upper bound on joint Renyi entropy

Renyi entropy of a random pair $(X,Y)$ with probability distribution $p_{X,Y}$ is defined by \begin{equation} H_\alpha(X,Y) = \frac{1}{1-\alpha}\log\sum_{x,y} p_{X,Y}(x,y)^\alpha. \end{equation} ...
4
votes
2answers
260 views

Maximum entropy priors in infinite dimensional spaces

Is there an extension of maximum entropy probability distributions for function spaces? For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as ...
0
votes
1answer
220 views

Entropy of inverse map for endomorphism case on surfaces

Hi, I know that in the diffeomorphism case the measure entropy of the T:M^{2}-->M^{2} (M smooth Rimannian surface) will be the same as the measure entropy of T^{-1}. But i need to know about the ...
3
votes
1answer
517 views

Convergence of Markov chains in terms of relative entropy

Consider a finite state, irreducible Markov chain with a rate matrix $Q$ and a stationary distribution $\pi$. Suppose the chain starts with the initial distribution $p$ at time $0$, then at time $t$ ...
3
votes
1answer
164 views

Ising entropy of a finite L_1 x L_2 lattice

We know the entropy per site of the 2-d Ising model from Onsager's solution. Has anybody also calculated the entropy for a finite rectangle of size L_1 x L_2 with periodic boundary conditions (i.e. on ...
5
votes
1answer
602 views

Bounding the entropy of a convolution

Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...
4
votes
2answers
522 views

Is there a simple relation between the entropy of a matrix and its characteristic polynomial?

Assume $M$ is an invertible positive matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues $\{ \lambda_n \}$ is $H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$. In ...
1
vote
0answers
136 views

Entropy of factors of Bernoulli schemes

Let $X$ be a Bernoulli scheme. A factor $\psi: X \to Y$ is finitary if for almost every $x \in X$ there exist integers $m \leq n$ such that the zero coordinates of $\psi(x)$ and $\psi(x')$ agree for ...
1
vote
0answers
55 views

Information amount of fuzzy data transfer [closed]

Suppose we have binary channel from which we are able to receive zeroes and ones. We also know apriory probability $p$ of receiving "1". Then we can calculate information amount of each digit $q$ we ...
10
votes
2answers
1k views

Proving a messy inequality

EDIT: After much work I was able to reduce the inequality to a single variable function which I need to show is non-positive. That function is (for $0\leq p\leq\frac{1}{2}$) $$\frac{p^2(\log(p))^2 - ...
2
votes
0answers
281 views

Does this inequality of negative relative entropy and quantum relative entropy hold?

Hello, everyone! Question I have a question about the relationship between general relative entropy and general quantum relative entropy: Given a unit vector $|i\rangle$ and two Hermitian matrices ...