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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28
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2answers
317 views

Information-theoretic aspects of mathematical systems?

It occured to me that when you perform division in some algebraic system, such as $\frac a b = c$ in $\mathbb R$, the division itself represents a relation of sorts between $a$ and $b$, and once you ...
22
votes
2answers
1k views

An information theory inequality which relates to Shannon Entropy

For $a_1,...,a_n,b_1,...,b_n>0,\quad$ define $a:=\sum a_i,\ b:=\sum b_i,\ s:=\sum \sqrt{a_ib_i}$. Is the following inequality true?: $${\frac{\Bigl(\prod a_i^{a_i}\Bigr)^\frac1a}a \cdot ...
16
votes
7answers
2k views

Intuitive explanation of entropy?

I have bumped many times into entropy, but it has never been clear for me why we use this formula: If $X$ is random variable then it's entropy is: $$H(X) = -\displaystyle\sum_{x} p(x)\log p(x).$$ ...
15
votes
3answers
2k views

Expanding and understanding the poison pills riddle

You might have heard of the riddle that asks you to identify one fake pill (poisoned) among 12 pills of identical appearance, with the fake pill being either lighter or heavier than the others. You ...
14
votes
3answers
1k views

How to tell if a code is lossless

Consider the following code mapping: $$a \mapsto 010, \quad b\mapsto 001, \quad c\mapsto 01$$ It's easy to see that the code isn't lossless by observing the code $01001$, which can be translated to ...
12
votes
2answers
354 views

metric in the Wasserstein space of gaussian measures

I am reading the paper "Wasserstein Geometry of Gaussian measures" by Asuka Takatsu (section 3 is of interest to me) and I have difficulties understanding how the metric is used. In particular, I am ...
11
votes
3answers
271 views

What is necessary to exchange messages between aliens? [closed]

Lets assume that two extreme intelligent species in the universe can exchange morse code messages for the first time. A can send messages to B and B to A, both have unlimited time, but they can not ...
11
votes
3answers
492 views

Has error correction been “solved”?

I recently came across Dan Piponi's blog post An End to Coding Theory and it left me very confused. The relevant portion is: But in the sixties Robert Gallager looked at generating random sparse ...
11
votes
2answers
843 views

Why is “h” used for entropy?

Why is the letter "h" (or "H") used to denote entropy in information theory, ergodic theory, and physics (and possibly other places)? Edit: I'm looking for an explanation of the original use of "H". ...
10
votes
3answers
371 views

What is the least amount of questions to find out the number that a person is thinking between 1 to 1000 when they are allowed to lie at most once

A person is thinking of a number between 1 and 1000. What is the least number of yes/no questions that we can ask and know what that person's number is given that the person is allowed to lie on at ...
9
votes
2answers
691 views

What is the relationship between the Boltzmann distribution and information theory?

I'm reading a paper on Boltzmann machines (a type of neural network in Machine Learning), and it mentions that "The Boltzmann distribution has some beautiful mathematical properties and it is ...
9
votes
2answers
691 views

In what sense is the Jeffreys prior invariant?

I've been trying to understand the motivation for the use of the Jeffreys prior in Bayesian statistics. Most texts I've read online make some comment to the effect that the Jeffreys prior is ...
8
votes
2answers
678 views

How to make the encoding of symbols needs only 1.58496 bits/symbol as carried out in theory?

I'm reading the tutorial of Information Gain, and I see the following page: I know in the example above, I can encode this way: A 0 B 10 C 11 and then this ...
8
votes
1answer
214 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 ...
7
votes
4answers
206 views

Why do lower probability messages contain more information?

This question comes from here. Suppose messages $m_1, m_2, \ldots$ can be sent (through a channel) to a receiver with probabilities $p_1, p_2, \ldots$. The amount of information transferred when a ...
7
votes
3answers
143 views

Using decimals of $\pi$ to store data

I read recently about an idea to, instead of storing actual data, converting the data to a string of digits and then store the index of where this pattern occurs in some number, for example $\pi$. The ...
7
votes
2answers
334 views

Can the entropy of a random variable with countably many outcomes be infinite?

Consider a random variable $X$ taking values over $\mathbb{N}$. Let $\mathbb{P}(X = i) = p_i$ for $i \in \mathbb{N}$. The entropy of $X$ is defined by $$H(X) = \sum_i -p_i \log p_i.$$ Is it possible ...
7
votes
3answers
198 views

Measure of how much information is lost in an implication

In an implication like $p \implies q$, is there some measure of how much information is lost in the implication? For example, consider the following implications, where $x \in \{0,1,\ldots,9\}$: ...
7
votes
1answer
259 views

Does “50/50 chance of.. . ” convey information?

I distinctly remember the professor in the undergrad introductory systems & control course saying that "when weather forecasters say there's a 50% chance of precipitation, they are conveying no ...
7
votes
1answer
401 views

Empirical distribution vs. the true one: How fast $KL( \hat{P}_n || Q)$ converges to $KL( P || Q)$?

Let $X_1,X_2,\dots$ be i.i.d. samples drawn from a discrete space $\mathcal{X}$ according to probability distribution $P$, and denote the resulting empirical distribution based on n samples by ...
7
votes
0answers
218 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
6
votes
3answers
336 views

Another Information Theory Riddle

The following nice riddle is a quote from the excellent, free-to-download book: Information Theory, Inference, and Learning Algorithms, written by David J.C. MacKay. In a magic trick, there are ...
6
votes
2answers
1k views

What is the trellis diagram for a linear block code?

For the convolutional codes there is so-called trellis diagram, for which the definition is rather clear for me, however in mathematical sense is not. I have heard that it can be defined for linear ...
6
votes
3answers
314 views

What is information theoretic entropy and its physical significance?

I have learned entropy in my information theory classes. The definition I got from text books was the average information content in a message sequence etc. But in one of the MIT videos related to ...
6
votes
4answers
309 views

Is probability objective?

As we know, probability is a measure of events. However, is it an objectively attribute of events, or just an illusion in ones' mind? For example, suppose that there is an empty black box with an ...
6
votes
2answers
197 views

What is the connectivity between Boltzmann's entropy expression and Shannon's entropy expression?

What is the connection between Boltzmann's entropy expression and Shannon's entropy expression? Shannon's entropy expression: $$ S= -K\sum_{i=1}^np_i\log (p_i) $$
6
votes
1answer
240 views

measure of information

We know that $l_i=\log \frac{1}{p_i}$ is the solution to the Shannon's source compression problem: $\arg \min_{\{l_i\}} \sum p_i l_i$ where the minimization is over all possible code length ...
6
votes
1answer
190 views

Lemma in Petersen's *Ergodic Theory*

I'm trying to understand the proof of Lemma 6.2.1 (p.260-261) in Petersen's Ergodic Theory. Specifically, I don't understand why $B_{n}^{A} \in \mathscr{B}(T^{-1}\alpha \vee \dots \vee T^{-n}\alpha)$ ...
6
votes
1answer
255 views

Lower bound on binomial coefficient

I encountered the following claim $$\frac{1}{n+1}2^{nH_2(k/n)} \le \binom{n}{k} \le 2^{nH_2(k/n)}$$ where $H_2$ is the binary entropy function. The upper bound is rather well known but how does one ...
5
votes
3answers
772 views

Inverse of binary entropy function for $0 \le x \le \frac{1}{2}$

I'm trying to find the inverse of $H_2(x) = -x \log_2 x - (1-x) \log_2 (1-x)$[1] subject to $0 \le x \le \frac{1}{2}$. This is for a computation, so an approximation is good enough. My approach was ...
5
votes
4answers
396 views

Increasing the number of repetitions decreases the error probability

In coding theory when we encode 101 with 111000111 we have certain error probability. how can one prove that increasing the number of repetitions decreases the error probability. Let the probability ...
5
votes
3answers
332 views

What does the -log[P(X)] mean in the calculation of entropy?

The entropy (self information) of a discrete random variable X is calculated as: $$ H(x)=E(-log[P(X)]) $$ What does the -log[P(X)] mean? It seems to be something like ""the self information of each ...
5
votes
1answer
662 views

Kullback-Leibler divergence based kernel

I'm looking to paper "A Kullback-Leibler Divergence Based Kernel for SVM Classification in Multimedia Applications". Author suggest to use kernel function for two distributions $p$ and $q$: $k(p,q)= ...
5
votes
2answers
135 views

How to Count the number of words over an alphabet subject to restrictions on letter count?

For an alphabet $X$, is there a method of computing how many words over $X$ of length $n$ there are where the number of occurrences of each letter must satisfy a system of equations? For example if ...
5
votes
2answers
292 views

Theoretical basis for overfitting

There are many examples in which making more "precise" predictions gives worse performance (e.g. Runge's phenomenon). My professor implied that there was a sound basis for choosing "simple" functions ...
5
votes
1answer
139 views

Card drawing algorithm

I want to know whether there is an algorithm for randomly and securely drawing cards from a deck. I was thinking about a way to play deck-based games online with no trusted party and no way to cheat. ...
5
votes
1answer
1k views

Proof of Pinsker's inequality.

How to prove the following known (Pinsker's) inequality? For two strictly positive sequences $(p_i)^n_{i=l}$ and $(q_i)^n_{i=l}$ with $\sum_{i=1}^np_i=\sum_{i=1}^nq_i=1$ one has ...
5
votes
1answer
53 views

Prove that bitstrings with 1/0-ratio different from 50/50 are compressable

I'm looking for a proof, that $$ \sum_{i=0}^{\lambda n} \binom{n}{i} \le 2^{nH(\lambda)} $$ with $n>0$, $0 \le \lambda \le 1/2$ and $ H(\lambda)=-[\lambda log \lambda + (1-\lambda) log (1-\lambda)] ...
5
votes
1answer
163 views

Entropy of the induced transformation

I need help with this problem: Let $(X,\mathcal{B},\mu,T)$ be a ergodic dynamical system in the probability space $(X,\mathcal{B},\mu)$. Let $A \in \mathcal{B}$ with $\mu(A)>0$. We define the ...
5
votes
0answers
62 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 ...
5
votes
1answer
159 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
votes
3answers
162 views

simulating a fair random process with an unfair one.

Let's say I have a stochastic process that outputs $1$ or $0$ with probability $p$ or $1-p$ respectively, $p\neq 1/2$. Let's assume this is a repeatable iid process. So I can generate $X_1,X_2\dots$ ...
4
votes
3answers
178 views

Estimating the entropy

Given a discrete random variable $X$, I would like to estimate the entropy of $Y=f(X)$ by sampling. I can sample uniformly from $X$. The samples are just random vectors of length $n$ where the entries ...
4
votes
3answers
132 views

Does any error correction code still work in such situation?

I'm looking for a kind of error correction code or solution that can correct my codeword in this case: My message holds k bits, and 2*k bits codeword (rate is 1/2) is produced by the generator ...
4
votes
1answer
130 views

Source coding theorem - optimum number of bits?

The source coding theorem says that information transfer with variable length code uses less bits and is equal to the entropy of the distribution. It also says that there is no code that uses lesser ...
4
votes
1answer
92 views

Shannon's MTC as 'information theory'

I'm a little confused as to whether or not this question belongs here or on http://cstheory.stackexchange.com/, so please, bear with me. I've been reading a few books on the concept of information, ...
4
votes
1answer
34 views

Applications of information theory in economics?

What are some direct applications of information theory in economics theory and/or finance? Any relevant articles, surveys, or book references are appreciated (especially if they are targeted to ...
4
votes
2answers
58 views

Determinant of Fisher information

In information geometry, the determinant of the Fisher information matrix is a natural volume form on a statistical manifold, so it has a nice geometrical interpretation. But what is it in ...
4
votes
1answer
94 views

Relationship between two measures of inequality

Let $A = \{a_1,\dots,a_n\}$ be a multi-set and let $B$ be the set of distinct elements in $A$. Now define $H(A) = -\frac1n \sum_{x \in B} f(x) \log_2(f(x)/n)$ where $f(x)$ is the number of times $x$ ...
4
votes
1answer
542 views

Upper bound for variance of an (arbitrary) zero-mean random variable $X$ given distance between it and a known random variable $Y$

I have a zero-mean Gaussian random variable $Y\sim\mathcal{N}(0,\sigma^2_X)$ with known variance $\sigma_X^2$. I also have a zero-mean random variable $X$, which may be dependent on $Y$ (though, I ...