The entropy tag has no usage guidance.

**1**

vote

**0**answers

35 views

### Existence of a proper entropy scaling for a discrete measure

In this (quite elementary) paper, the scaled entropy of an unbounded measure $\mu$ on $\mathbb{N}$ is defined by
$$
h_c(\mu) := \lim_{\epsilon \to 0} \limsup_{n \to \infty} \dfrac{H^\epsilon(X_n)}{c(...

**1**

vote

**0**answers

67 views

### How to prove this Gaussian Mixture theorem? (Fitting/Overfitting)

Note from OP: I gave up and reposted this Question with a Bounty on Cross Validated HERE.
In certain applications, we approximate an unknown pdf by placing uniformly weighted Gaussian terms at each ...

**1**

vote

**2**answers

61 views

### Reference for the monotonicity in $\alpha$ of the Rényi entropy

I'd like to have a reference for the property $0 \leq \alpha < \alpha' \leq \infty \implies R_\alpha(\mu) > R_{\alpha'}(\mu)$, where $R_\alpha(\mu)$ is the Rényi entropy of order $\alpha$ of a ...

**1**

vote

**1**answer

43 views

### Maximizing joint entropy?

I'm stuck trying to find the maximum entropy probability distribution taking into account a joint distribution.
Basically, I want to find the maximum entropy expression for $p(x,y)$ when the marginal ...

**-1**

votes

**1**answer

85 views

### Proving maximal entropy [closed]

It is quite easy to prove that
$$H(S) \leq \log_2(|A|),$$
where $A$ is the number of events, using the Jensen inequality
$$H(S) = E_S[\log_2(\frac{1}{P_S(s)})]\leq \log_2(E_S[(\frac{1}{P_S(s)})]) =...

**2**

votes

**1**answer

123 views

### About Renyi entropy

If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = \frac{...

**3**

votes

**0**answers

80 views

### An inequality involving conditional variance and its connection to information theory

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$ ...

**2**

votes

**0**answers

37 views

### TAP expression for entropy [closed]

This paper by Barton and Cocco: http://www.phys.ens.fr/~cocco/Art/articlejstat.pdf
claims on page 17 (Formula (30)) an expression for the "high-temperature" entropy of an Ising model, given its ...

**1**

vote

**0**answers

98 views

### weak-* versus entropy growth

General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...

**1**

vote

**0**answers

95 views

### Strict factor of a dynamical system with the same entropy [closed]

Say that a factor of an invertible measure-preserving transformation $T$ is strict if it is not isomorphic to $T$. Does there exist an invertibe mpt $T$ such that $0 < h(T) < \infty$ and ...

**6**

votes

**1**answer

240 views

### Finding discrete entropy via differential entropy

In a recent math.se question the following was asked which I have slightly edited.
"
Consider a fixed and given $n$ by $n$ matrix $M$ whose elements are chosen from $\{-1,1\}$. Consider also a random ...

**0**

votes

**1**answer

117 views

### entropy growth of invariant measures - General question

In general, given a sequence of shift-invariant measures $\eta_{n}$ on $\{0,1\}^{\mathbb{N}}$ what to do to guarantee this convergence of entropies: $$h(\eta_{n}) \rightarrow \log2?$$
Because I'm ...

**1**

vote

**1**answer

119 views

### entropy and d-bar: how do we estimate continuity?

Let $G = \{0,1\}^{\mathbb{N}} = \mathbb{Z}_{2}^{\mathbb{N}}$ be the Bernoulli space of two symbols, let $\sigma$ be the shift map and $M(G)$ the set of $\sigma$-invariant probabilities. Let $\bar{d}$ ...

**5**

votes

**0**answers

72 views

### Basis functions for approximation of a convex function on unit simplex

Consider the unit $D$-simplex $S^D=\left\lbrace (x_0, x_1, \ldots, x_D) \in \mathbb{R}^{D+1} \mid \sum\limits_{i=0}^{D}x_i = 1, x_i \geq 0 \right\rbrace$. I have a bounded, convex function $f:S^D\to\...

**4**

votes

**2**answers

270 views

### Entropy equals zero?

Imagine you have a shift invariant ($\sigma$-invariant) probability measure $\eta$
in the Bernoulli space $\{0,1\}^{\mathbb{N}}$. Define
$\mathcal{P} = \{[0],[1]\}$;
$\mathcal{P}^{n} = \mathcal{P}\...

**7**

votes

**2**answers

221 views

### Estimating entropy conditional to an event

Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$.
Does there exists a constant $K>0$...

**0**

votes

**1**answer

148 views

### joining or coupling

given two shift invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$, is there a way to construct joinings of them? It's very diffcult, in general, to find exactly the minimal joining i.e, ...

**4**

votes

**1**answer

255 views

### Do binary symmetric channels maximize mutual information?

Consider the following setup: $(X, Y)$ is a doubly symmetric binary source with parameter $0 < p < 1/2$, i.e., $X \sim \text{Bernoulli}(1/2)$, $Z \sim \text{Bernoulli}(p)$ and $Y = X \oplus Z$. ...

**0**

votes

**0**answers

91 views

### Entropy, convergence and invariant measures

Could you give conditions that a sequence of shift invariant measures $\eta_{n}$ has to satisfy in order to happen this convergence in terms of entropies $h(\eta_{n})$:
$h(\eta_{n}) \rightarrow \log2$...

**1**

vote

**1**answer

177 views

### Entropy, Convergence

imagine you have a sequence $\eta_{n}$ of (shift) invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ that satisfy the following: there are a $0<\delta <1$ and an $N$ such that $$n &...

**3**

votes

**1**answer

270 views

### Information theory from negative probability

Szekely provides a convincing argument of negative probability here:
http://www.wilmott.com/pdfs/100609_gjs.pdf
What does a reformulation of classical information theory built from negative ...

**7**

votes

**2**answers

362 views

### Maximal entropy distribution with given conditionals

It is well known that of all the joint distributions $p(x,y)$ with fixed marginals $p(x),p(y)$, the one with the highest entropy is:
$$
p(x,y)=p(x)p(y).
$$
Suppose instead that we have conditionals. ...

**0**

votes

**1**answer

84 views

### Asymptotically full stationary process

Let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is asymptotically full if for every increasing sequence of subsets $B_n \subset A^n$ such that $\dfrac{\#B_n}{\#...

**3**

votes

**1**answer

257 views

### Convolution of measures - entropy growth

Imagine you have two shift-invariant measures $\mu, \nu$ in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ with positive entropy and both are not the Bernoulli measure $(\frac{1}{2},\frac{1}{2})$. I know ...

**3**

votes

**3**answers

225 views

### Maximizing entropy under constraints

This question is about an extension of the variational principle in thermodynamical formalism when one adds linear constraints to the measures.
Consider the one-sided shift $\sigma:\mathcal{A}^\...

**2**

votes

**2**answers

279 views

### Entropy inequality

Let $P,Q$ be probabilities on a finite set $A$ with $Q(a)\gt 0$, for all $a\in A$, and let $H(P)$, $H(Q)$ denote their entropies and $D(P\,\|\,Q)$ denote their Kullback-Leibler distance.
Is it always ...

**1**

vote

**0**answers

115 views

### Estimates of entropy of functional spaces

Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it.
...

**2**

votes

**1**answer

198 views

### Embeddings of subshifts

Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a ...

**9**

votes

**2**answers

485 views

### Expected centered entropy of the binomial distribution

In short, the function I am interested in is the following:
$$I_n(p) = \sum_{k=0}^n \binom{n}{k} p^k (1-p)^{n-k} \left[h(p) - h\left(\tfrac{k}{n}\right)\right],$$
where $h(x) \triangleq -x \log x - (1-...

**1**

vote

**1**answer

378 views

### Coupon Collector Problem for Non-Uniform Coupons: Bound on the number of missed Coupons

Suppose $\mathcal B=\{1,2,..,b\}$ is the set of all possible coupons, with $\mathbf p = ( p_1,p_2,...,p_b)$ assigning the probability of occurrence for all coupons in $\mathcal B$.
The "traditional ...

**2**

votes

**0**answers

98 views

### Concentration bound in high min entropy distribution

Let $(X_{1},\dots,X_{m})$ be joint distribution on $\{0,1\}^{m}$ with that $H_{\infty}(X_{1},\cdots,X_{m})\geq m-r$, where $H_{\infty}$ means min-entropy.
Let $P_{1},...,P_{n}\subseteq [m]$ be sets ...

**1**

vote

**1**answer

448 views

### Discrete Maximum Entropy Distribution with given mean

For a given mean $\mu$, what is the entropy maximizing probability distribution on the nonnegative integers?
Different sources indicated either the geometric or the Poisson distribution for this. As ...

**2**

votes

**0**answers

72 views

### Sequence transformations that are entropy invariant

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Define entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$
by $$H(\mathcal{A},m)=-\...

**2**

votes

**0**answers

248 views

### Interpretation of Shannon Entropy Application

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Let entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$ be given
by $$H(\mathcal{A}...

**1**

vote

**0**answers

44 views

### Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution $m\...

**1**

vote

**1**answer

111 views

### Limiting Entropy of deterministic sequences - 1

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.
Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at ...

**1**

vote

**1**answer

86 views

### Entropy difference dominance of sequences

Consider a collection of positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$.
Similarly for the collection $\{a_i\}_{i=1}^{m+1}$ form the distribution $q_i=\...

**1**

vote

**1**answer

61 views

### Entropy dominance of certain restricted sequenes

Say you have positive $\{a_i\}_{i=1}^n$ and you have $p_i=\frac{a_i}{\sum_{i=1}^na_i}$, then assume you have a $C$ such that $C<2a_n\ll\sum_{i=1}^na_i$ (that is $C$ is not very large), then define $...

**2**

votes

**1**answer

83 views

### Entropy dominance

Let $0<a<b<c$ be distinct positive reals.
Define four different probability distributions:
$$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$
$$\mathcal{P}_{bc}:P_{b,bc}=\frac{b}{b+c}=1-...

**8**

votes

**1**answer

237 views

### Higher moments of information and Renyi entropy

For a given discrete probability distribution, Shannon entropy can be though as an expectation value $\langle - \log p \rangle$ (see also: What is entropy, really?, What is the role of the logarithm ...

**2**

votes

**0**answers

106 views

### Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measure on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as
$$D(Q\|P)=\int_X \log\left(\frac{dQ}{dP}\...

**17**

votes

**8**answers

2k views

### When do people actually use the maximum entropy distribution?

One of the standard problems in convex optimization is the calculation of the maximum entropy distribution that satisfies some set of criteria. For example, if $\mathbf{x} \in \mathbb R^n$ is an ...

**3**

votes

**1**answer

143 views

### Average entropy of quantum system in bipartite pure state for finite temperature

[I got halfway through writing this when I found the paper that answers the question in (essentially) the affirmative. I'll post it anyways in case anyone is interested.]
Background: If a random ...

**2**

votes

**1**answer

188 views

### Which is the right way to compute the Approximate Entropy (ApEn)?

My problem is the inconsistency between the definition and the computation of the Approximate entropy (ApEn).
Suppose $u = (u_i:1\leq i \leq N)$ is a sequence of ...

**1**

vote

**1**answer

120 views

### Entropy on a draw from a random distribution.

Suppose I am attempting to calculate the entropy of a continuous, normally distributed random variable $X$, from the distribution $\mathcal{N}(\mu, \sigma)$. This is easy to to do - I just calculate
$...

**4**

votes

**0**answers

102 views

### $q$-deformations of fundamental equation of information and entropies

Classical information theory: fundamental equation of information
In classical information theory, the information $I(A)$ of an event $A$ (any element of the $\sigma$-algebra $\mathcal F$ of a given ...

**1**

vote

**0**answers

57 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, ...

**1**

vote

**1**answer

190 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 ...

**3**

votes

**0**answers

129 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

**1**answer

130 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 ...