The science of compressing and communicating information. It is a branch of applied mathematics and electrical engineering. Though originally the focus was on digital communications and computing, it now finds wide use in biology, physics and other sciences.

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7
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173 views

Error correction code handling deletions and insertions

I have data which is expressed in form of fixed-length sequence of decimal digits, typically 10 digits in length. I'm looking for error correction code that would allow me to append one or more ...
4
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37 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 ...
4
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71 views

Dividing a deck of cards using only imagination

The idea came up from a discussion I had with my friends. Suppose we want to play a game using a deck of cards, and we can't use any physical materials. If we are intelligent enough, we can remember ...
4
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0answers
96 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le ...
4
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249 views

Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis. Correlation of the series changes over time and across ...
4
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282 views

Inequalities involving the probability density function and variance

I am wondering whether anyone knows of any any inequalities involving the probability density function of an unknown distribution (as opposed to the cumulative distribution function) and its known ...
3
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39 views

Prove that communication protocol complexity less than $n\epsilon$

Alice and Bob get as an input words $x$ and $y$, which consist of $0$ and $1$. Length of $x$ is $n$ and length of $y$ is $2n$. They want to know if the word $x$ is subword of word $y$. For example, ...
3
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104 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 ...
3
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50 views

Is there a conditional version of the asymptotic equipartition property?

Let $X_i$ be independent random variables with $\operatorname{Pr}(X_i = x) = p_x$, and let $F_n$ be the empirical frequency distribution of $X_1, \ldots, X_n$: that is, $(F_n)_x$ For any frequency ...
3
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87 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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222 views

Intuition for Fisher information metric

In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\dots,\theta_n)$, the Riemaanian metric usually defined is the Fisher information metric $$g_{ij}(\partial_i,\partial_j)=\int \partial_i(\log ...
3
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67 views

Calculate sums of logs in precision

I am encountering a situation where I cannot calculate exact sum of a seris of logorithms in calculating entropy. Suppose we have a series of numbers $p_i$ and we want to calculate $\sum_ilog(p_i)$, ...
3
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212 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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0answers
29 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 ...
2
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31 views

A question regarding binomial coefficient

This question arose during solving an information theory problem. Suppose $l$ is the smallest integer such that $$2^l\geq {n\choose k}$$ define $\rho=\frac{k}{n}$. How we can characterize $\rho$ as a ...
2
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40 views

Finding the mean and the variance of a martingale using concentration inequalities

I am trying to find the mean and the variance of a martingale defined as the maximized likelihood ratios over some finite parameter space. The way I want to do this is through Azuma's inequality (or ...
2
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0answers
34 views

Kullback–Leibler divergence with elements that are $0$

I have a problem that I need to calculate Kullback–Leibler divergence, but the problem is that I have some elements that are $0$. Is there a way that I'm able to deal with situations like that? I know ...
2
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31 views

Upper bound of mutual information in a Markov chain

Consider binary random variables $X$ and $V$ with marginal distributions $p$ and $\pi$ respectively and also the conditional distribution $p(X=x\mid V=v)=q(x\mid v)$, where $x\in\{-b,b\}$ and ...
2
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22 views

Fastest decaying probabilities with infinite entropy

A well known theorem of analysis is that there is no slowest rate of divergence of a series. "Completely irrelevantly," we know that there exists probabilities (e.g. $\mathtt{Const}/(n\log^2 ...
2
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0answers
66 views

Relation between entropy of variable and entropy of conditioned variable

Let $X$ be a discrete random variable, and let $E$ be an event on the same probability space as $X$. Let $X_E$ be $X$ conditioned on the event $E$. Is there a general relationship between the Shannon ...
2
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30 views

Is it true that $\sum _{i=0}^a (q-1)^i\binom {n}{i} \leq q^{H_q(a/n)n}$?

Given $q \in \mathbb N$, $q\geq 2$ is it true that \begin{equation*} \sum _{i=0}^a (q-1)^i\binom {n}{i} \leq q^{H_q(a/n)n}? \end{equation*} Here $H_q(x) = x\log _q(1/x) + (1-x)\log _q(1/(1-x))$ is the ...
2
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82 views

Explanation of Radon-Nikodym derivates wrt to probabilities

I am currently working in communications, where a lot of work is done via probability calculations (densities and such). As I am not a mathematician, I do have a quite hard time understanding one ...
2
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115 views

Information content of universal sentence

What is the information content of a sentence S like 'one has a successor'. To me, it looks like if we assume no a priori knowledge, both S and it's negation will have equal probablity 1/2. This is ...
2
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114 views

Joint distribution between a uniform random variable and a function which is “almost” independent from it

Motivation Let $f(\cdot)$ be a (possibly randomized) function, such that for any random variable $X$ (taking values from a finite set $D$), $X$ and $f(X)$ are statistically independent. Let $U, U_1, ...
2
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121 views

Non-zero Conditional Differential Entropy between a random variable and a function of it

Let two continuous random variables, where the one is a function of the other: $X\, $ and $\, Y=g\left(X\right)$. Their mutual information is defined as ...
2
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103 views

Lower bound on uncertainty reduction

Let $T$ be a set of tuples such that each score tuple $s(t_i)$, $t_i \in T$ is uncertain (i.e., not known deterministically). The score $s(t_i)$ can be represented as a uniform probability density ...
2
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52 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 ...
2
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0answers
76 views

Simple / Fast way to calculate the period of a arbitrary polynomial?

If I get a polynomial, say $u^5 + u^4 + u^2 + 1$, what is a simple and fast (by which I mean not to much to write) way to get the period of this term? Should I first test whether the polynomial is ...
2
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60 views

Proving Symmetrized Kullback-Leibler divergence

Kullback in his "Information theory and statistics" gives the symmetrized divergence as follows $J(1,2)=\iint(f(x,y)-g(x)h(y))log{\frac{f(x,y)}{g(x)h(y)}}$ Later (p.8), he states that symmetrized ...
2
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146 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 ...
2
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55 views

is there a computationally efficient formula for computing the mutual information between two continuous variables?

I need to compute the mutual information between two continuous variables. Below is an equation shown to compute the mutual information between a variable $X$ and $Y$. $I(X;Y) = \int_Y \int_X ...
1
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0answers
12 views

Are measures of information model specific?

Does an information measure for a signal do a better job if it assumes some things about the signal? For example: I have a digital stream of data, 0s and 1s coming at a clock rate $r$. What is the ...
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73 views

What do two number on top of each other in square brackets mean?

Im currently going through "Universal Portfolios with Side Information" by Cover and Ordentlich [96]. Near the end of the paper, they provide a formula for calculating weights of a Universal Portfolio ...
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14 views

Encoder based on large similar data

Let us say you (Alice) and another agent (Bob) share a large piece of data (say, the Gutenberg project collection of books, or the Linux kernel. You want to send a smaller but still large piece of ...
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327 views

Calculating minimum hamming distance of a code

We use hamming code of (7,4,3); Given 4 bits of information, we'll add 3 bits of parity, and one more parity bit for the 7-bits code. Given $x_3,x_5,x_6,x_7$ $x_1 = (x_3+x_5+x_7) \mod 2$ $x_2 = ...
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44 views

Using mutual information to estimate correlation between a continuous variable and a categorical variable

As for the title, the idea is to use mutual information, here and after MI, to estimate "correlation" (defined as "how much I know about A when I know B") between a continuous variable and a ...
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0answers
50 views

Continous wavelet transform and shannon Entropy.

Note: I have asked the same question on signal processing forum,but didn't get any answer. so it might be more like a math or physics question. Hope you don't consider it as cross-post. I am trying to ...
1
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0answers
29 views

Information Entropy Applied to Complexity Theory

I was just wondering whether or not information entropy has significant applications to complexity theory. I ask because of a simple example I thought of. It comes from a riddle. Suppose you had 8 ...
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90 views

Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...
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50 views

Negative exponential/ exponential power distribution between 0 .0 and 1.0?

Note: I'm not very familiar with distribution and higher level math Heyho, I'm currently looking for a way to generate random values between 0.0 and 1.0 with an exponential power or negative ...
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0answers
29 views

How to define a utility of an information source?

This is a more specific (and, hopefully, clearer version of a previous question). The utility of discovering the value of a random variable $X$ can be defined to be its information content. When ...
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0answers
24 views

Does this quantity have a meaning or relation to something else?

$X$ is a discrete random variable taking values in $\{0,1,2,3,\ldots\}$ with a probability mass function $p_X(n)$. Let $$U_k(X)=\sum_{n=k}^\infty\sqrt{np_X(n)p_X(n-k)}$$ where $k$ is a positive ...
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30 views

Independent transformation of probability measures

I have a pair of dependent random variable $(\theta, X)$ where $\theta\in K$ for a compact subset $K\subset\mathbb{R}$ and $X\in\mathbb{R}^d$ with marginals $P_{\theta}$ and $P_X$. I want to ...
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17 views

Orthogonality of parameter space

Folks, I have a basic information theory question. I am fitting a highly parameterized model to some data. In general: $$ y = \sum\limits_{i=1}^{13} \alpha_i X_i $$ Currently I use gradient descent ...
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48 views

what is relative entropy between to random binary string with length of $L_1$ & $L_2$?

I want calculate relative entropy between two strings of binary such as: $L_1:11000100011101001$ $L_2:00101110110111001$ It is primarily when the lengths of two strings is same and in general when ...
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46 views

Rate Distortion function and similarity of probability distribution

I have been reading about the rate distortion function in which the fundamental limit of compression of a random variable $X$ to another random variable $Y$ taking values on smaller alphabet within ...
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111 views

How many code words if average code length equals entropy

I've been given a proof to show: 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 $q\equiv1 \mod{(r-1)}$. ...
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122 views

KL divergence of multinomial distribution

Consider $q(x)$ be a Multinomial distribution over $\{1, \ldots, k\}$ with parameters $\{\theta_1,\ldots, \theta_k\}$. And p(x) over $\{1,\ldots, k\}$ with distribution $p(x)=\frac{1}{k}$. Then what ...
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36 views

Calculating entropy of Naive Bayes random variables

Suppose a Naive Bayes graphical model with binary random variables is given by $$P(y,x_1,x_2,...,x_n)=P(y)P(x_1|y)...P(x_n|y)$$ Attempting to calculate $I(x_1,...,x_n;y)$ raises the question: how can ...
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64 views

Maximum of the expectation of a concave function

Let's have a function $f(x, \theta)$, and some probability distribution on $x$. Let's say I have found $\theta^* = \operatorname{argmax}(f(E[x], \theta) $, and $f$ is concave in $x$. I would like to ...