1
vote
1answer
76 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 ...
2
votes
1answer
79 views

Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Denote by $p(x)$ the input probability distribution. The mutual information between the input and the output of ...
7
votes
3answers
694 views

A conjecture about the entropy of matrix vector products

Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ ...
2
votes
0answers
83 views

Lower convex envelope of a function involving entropy

Suppose two discrete random variables $X$ and $Y$ defined on finite sets $\mathcal{X}$ and $\mathcal{Y}$ are given and also suppose the conditional distribution $P_{Y|X}$ (i.e, channel) is fixed. We ...
19
votes
0answers
623 views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
2
votes
1answer
164 views

Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$. Is it true that: If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...
0
votes
0answers
72 views

An attempt to solve “Maximization of a total variation distance subject to another total variation distance in Markov chain”

I have been trying to solve Maximization of a total variation distance subject to another total variation distance in Markov chain. As a recall, suppose we have a pair of correlated random variables ...
3
votes
0answers
173 views

Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
5
votes
0answers
183 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
131 views

An optimization in Markov Chain

We are given two correlated random variables $V$ and $X$ supported over a finite alphabets $\mathcal{V}$ and $\mathcal{X}$. Suppose the marginal $P_V$ and conditional distribution $P_{X|V}$ are ...
3
votes
1answer
370 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
2answers
303 views

What is the maximum entropy distribution on the natural numbers?

On the reals $\mathbb{R}$, the maximum entropy distribution with a given mean and variance is the Gaussian distribution. Let $\mu, \sigma > 0$. What is the maximum entropy distribution on the ...
16
votes
1answer
695 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 ...
1
vote
0answers
110 views

expected inverse of circulant plus random diagonal

I have a deterministic circulant matrix $R$ and a random diagonal matrix $X$ where all elements are IID and positive. I need to determine the expected inverse of $R+X$, that is: Evaluate, in closed, ...
0
votes
1answer
236 views

Size of KL-divergence neighbourhoods

I am new here. I was reading another post here and this got me wondering what can be said about the size of the following kl divergence neighborhoods. Consider these two kl-divergence neighbourhood ...
1
vote
1answer
163 views

Set of distributions that minimize KL divergence,

Assuming that $p,q$ are probability distributions defined on the same support $\{x_i\}_{0 \leq i \leq n}$, $\epsilon$ a small real number, and $D_{KL}$ the Kullback-Leibler divergence, is there a ...
2
votes
0answers
94 views

Information theoretic privacy and distance of probability measures!

I came across the notion of information theoretic privacy in the paper of Yamamoto ("A source coding problem for sources with additional outputs to keep secret from the receiver or wiretappers "). The ...
2
votes
0answers
91 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
95 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 ...
5
votes
1answer
87 views

a measure of difference for arrangements of sphere points

Suppose one has a distribution of $N$ points on the sphere. Is there an agreed upon metric for the difference of this distribution and $N$ equidistant points on the sphere? To me entropy seems like ...
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
457 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
0answers
169 views

Axioms for Mutual Information

I am interesting in axiomatic justifications for concepts in information theory. I have found many axiomatizations for Shannon's entropy and for the Kullback-Leibler divergence, as well as their ...
2
votes
1answer
264 views

A Johnson-Lindenstrauss lemma for finite fields?

Given $m$ points in $\mathbb{R}^N$, the Johnson-Lindenstrauss lemma guarantees the existence of a linear operator $\mathbb{R}^N\rightarrow\mathbb{R}^n$ that nearly preserves pairwise distances between ...
5
votes
0answers
166 views

Chernoff bound in the not-quite-sub-exponential case

In Terry Tao's notes on Concentration of measure, Exercise 7 indicates that the Chernoff bound can be generalized to sub-exponential random variables: ...
2
votes
1answer
180 views

KL divergence(s) comparison,

Hi, $P_1$, $P_2$, $P_3$ are probability distributions defined on the same support. Knowing that $H(P_1) < H(P_2) < H(P_3)$, can we compare $D_{KL}(P_2,P_1)$ and $D_{KL}(P_3,P_1)$ ? (H is the ...
1
vote
1answer
291 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 ...
2
votes
1answer
287 views

Derivative of a random process

Consider $w(t)$ as Guassian random process, with $w(t)$ being $\mathcal{N}(\mu,\sigma)$ and i.i.d for all t. I consider applying a (stochastic)derivative operation to the random process. What is the ...
8
votes
0answers
239 views

Uncertainty principle in Entropy terms

Math Questions: Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm $ ||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2}, $ and Fourier transform $ (F\psi)(\xi) = ...
2
votes
1answer
145 views

Moments of random matrices - when are they finite

I need to evaluate the moment $$\mathbb{E} (AX)^n,$$ where A is an NxN Hermitian square matrix, and X is $$X=ZZ^{\ast},$$ where $Z=\mu+Y$, where $\mu$ is mean of $Z$ and $Y$ is a zero-mean complex ...
2
votes
0answers
70 views

“Soft” Voronoi cells or statistical criterias

It is probably some basic statistics question, but... Informally 1: How to choose "criteria", such that it will guarantee that error decision probability is less than "epsilon", and maximize ...
3
votes
0answers
124 views

What distribution(s) of delays make(s) timing attacks hardest?

$H$ is (Shannon) entropy. In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 ...
5
votes
2answers
500 views

Sufficient Condition for Exponential Decay in Chernoff Bound (Large Deviations)

Let $X_i$ ($i=1,...,n$) be a sequence of independent and identically distributed random variables. Denote $\mu=\mathbb{E}[X_i]$ and $S_n=\frac{1}{n}\sum_{i=1}^nX_i$. This question concerns the tail ...
3
votes
1answer
244 views

convex combination of two covariance estimates

I am interested in covaraince matrix estimation. In brief: I have two estimates of the covariance matrix, and now I want to form a bona fide convex combination of the two. Background: I have studied ...
3
votes
0answers
134 views

Find a minimum entropy code for a simple gibbs random field.

Just to make precise what I am talking about, I will include the definition of a minimum entropy code. I will then define the precise markov random field I am asking about. In the rest of this ...
4
votes
2answers
358 views

Proving that a complicated function is eventually concave

I have a function $f:\mathbb{R}^+ \to \mathbb{R}^+$ that I want to prove is eventually concave - i.e. that there exists $\gamma _0 > 0$ such that for every $\gamma>\gamma_0$, $f(\gamma)$ is ...
4
votes
2answers
261 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
186 views

MMSE estimator expressed through cumulants

I have a linear model $$Y=HX+N,$$ where $H$ is a matrix and $X$ are drawn from $p_X(X)$, and $N$ is Gaussian noise variates. Now, if $X$ is multivariate Gaussian, then a linear estimator ...
3
votes
1answer
2k views

With Huffman code, why do we still need Shannon code?

I'm studying information theory by myself. I'm confused about that since we already have Huffman code, which is the optimal code method, why are Shannon code and some other code still useful? I ...
1
vote
1answer
146 views

The degrees in a random subgraph

Fix some positive integers $N$ and $d_k$, $k=1,2,\dots$ with $N=\sum_{k=1}^\infty d_k$. Suppose you have a graph $G$ taken randomly uniformly among the set of all (unoriented) graphs with $N$ ...
0
votes
2answers
209 views

multivariate distributions unaffected by unitary transformations

Hi, In my research I reached some very nice results for IID complex Gaussian vectors $\bf{x}$. Now I realize that my results hold for any random vectors that are unaffected by a unitary map, i.e., ...
6
votes
2answers
186 views

Number of neigbour Voronoi cells for a random set of points on S^k or cube [-1, 1]^k?

Consider $S^k \subset R^{k+1} $. Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ). Consider Voronoi cell around each point. How many neighbours would a cell have ...
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 ...
2
votes
1answer
206 views

Error bounds for truncating a probability distribution based on the entropy?

Heuristic Background Consider a set of states labeled $n=1,2,...$ in order of non-increasing probability $p(n)$. The standard Shannon argument gives meaning to the entropy $S$ of $p$ in terms of the ...
1
vote
2answers
281 views

Measuring the independence between the components of a stochastic process

In a context of blind source separation (e.g. you want to extract the voice of a singer from a song), many approaches consist in maximizing the independence between the components of a certain ...
1
vote
1answer
408 views

Bounding Entropy in terms of KL-Divergence

Let $h(X)$ be the differential entropy of a continuous random variable $X$ with density $f$, and let $Y$ be another continuous random variable with density $g$. If $KL(X\mid\mid Y)$ is the ...
3
votes
0answers
94 views

Is a parametric family which is universally consistent for multiple quantiles impossible?

Suppose I am dead-set on using Bayesian inference on independent and identically distributed data, but I'm lazy and insist on using a parametric likelihood function come what may. I'd be reassured to ...
2
votes
1answer
437 views

Divergence between two random variables

I have two Gaussian random variables $X$ and $Y$, each of which is an estimator of an underlying quantity. I need to measure whether $Y$ is estimating something different than $X$. So if the mean of ...
2
votes
0answers
125 views

finding rank-3 tensors compatible with a rank-2 tensor projection

I am interested in the following problem: Consider a rank-3 symmetric tensor $\boldsymbol{\sigma}$ with $\sigma_{ijk}$ where $\sigma_{ijk}$ can be 0 or 1, and the symmetry is with respect to any ...