The entropy tag has no usage guidance.

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110 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 ...

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36 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 ...

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197 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 ...

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**2**answers

225 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,
...

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56 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 ...

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238 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 ...

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219 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 ...

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455 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 ...

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**1**answer

369 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**

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**1**answer

410 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 ...

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**1**answer

207 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 ...

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1k 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 ...

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**1**answer

132 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 ...

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148 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 ...

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117 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 ...

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161 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 ...

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98 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 ...

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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 ...

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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 ...

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588 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 $S_i^j:=\sum_{k=i}^j x_k$. ...

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2k 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 ...

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116 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 ...

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700 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 ...

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587 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 ...

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**1**answer

542 views

### Calculate channel capacity of general channel under constraint

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 corresponds ...

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418 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?

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174 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} ...

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2k 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, ...

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135 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 ...

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256 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 ...

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431 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}
...

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293 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 ...

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222 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 ...

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**1**answer

856 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 $t$ ...

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**1**answer

183 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 ...

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**1**answer

812 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 ...

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1k 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 ...

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**0**answers

165 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 ...

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59 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 ...

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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 - ...

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313 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 ...

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376 views

### Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...

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256 views

### Incremental computation of a conditional entropy

Is it possible to compute a conditional entropy (see the two following formulas) in an incremental manner ? That is, the sets C and K are not fix: each time we have a new element c, set K may increase ...

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153 views

### asymptotic behaviour of the entropy and degeneracy

For each $n \in \mathbb{N}$ let $X_n$ be a random variable taking its values in a finite set $E_n$ with $P(X_n=x_n)>0$ for all $x_n \in E_n$. Say that $X_n$ is asymptotically degenerate if ...

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576 views

### Low degree polynomial approximation for the entropy function

Let $X$ be a discrete random variable with possible values
$\{x_1,\ldots,x_n\}$, and let $p$ denote the probability mass function of
$X$. In addition, denote $p_i=p(x_i)$.
The entropy of $X$ is ...

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**1**answer

384 views

### existence of finitely additive measures with zero entropy

Let $X$ be a countable set and $\mathcal M(X)$ be the set of finitely additive probability measures on $X$. If $\mu\in\mathcal M(X)$, I define the entropy of $\mu$ to be
$$
...

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248 views

### Compact group extension of a zero entropy system.

Suppose $T: X \to X$ is a continuous map
and $\mu$ a $T$-ergodic probability measure over the
Borel sets of $X$.
Now, suppose $K \subset \mathrm{Hom}(X)$ is a compact group
of measure-preserving ...

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**1**answer

1k views

### System with invariant measure, but no ergodic measure.

Question
Examples of continuous transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$).
Notice ...

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804 views

### Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures.

Cases where
$sup_{\mu \in E(T)} h_\mu(T)
\neq
\sup_{\mu \in M(T)} h_\mu(T)$.
Background
For a topological space $X$,
let $T: X \to X$ be a continuous application.
Then, call the set of ...

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**1**answer

412 views

### closed form expression for Rényi entropy for multivariate Gaussian distributions

Is there any closed form expression for Rényi entropy of a set variables with multivariate Gaussian distribution?