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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Bound the entropy knowing the largest denominator

The problem comes up from considering sampling from a discrete set of $n$ items with integer weights. The $i$th item has weight $w_i$, the probability getting chosen is $w_i/\sum_j w_j$. Certainly the ...
4
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2answers
37 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 ...
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1answer
25 views

$H(X\mid Y_1, Y_2) \leq H(X\mid Y_1)?$ (Conditional Entropy with conditioning on multiple RVs)

In short, my question is whether the "conditioning reduces entropy" maxim is also true when conditioning on one random variable as compared to conditioning on two: $$H(X\mid Y_1, Y_2) \leq H(X\mid ...
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20 views

Hypercontractivity of Markov Operator

I have been reading a paper by Ahlswede and Gacs on hypercontractivity of Markov operator (see here 1) and its application to information theory. To be honest, I could not fully understand the ...
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1answer
45 views

Is it possible to study information theory while studying a first course on probability?

I'm currently taking a course on intro to probability. The course is not mathematically rigorous and does not invoke theorems from real analysis, etc. The course covers all the way from basic ...
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23 views

Self-information, one event half as likely than another event conveys twice the amount of information?

I was reading the following: "If one event is half as likely as another, then learning about the former event shouldconvey twice as much information as the latter" I know it should be easy to ...
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1answer
45 views

How is this paper using probability notation?

I am trying to understand this paper about documents and sentences. At the end of page three, they say: Let g(wi, wj ) be the distance between two events (1 if in the same sentence, 2 in neighboring, ...
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28 views

A simple formula for pseudo-random binary string

I'm trying to illustrate the concept of Kolmogorov complexity by comparing two seemingly-random binary strings: while one is truly random, the other can be created by a program smaller than its ...
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1answer
33 views

Erasure Codes with Simplex Locality

In here, theorem $1.1$. there is this line 'Since $G$ has full rank it is possible to enlarge $N$ to a set $N^{'}$ ... exactly $k-1$. Note that the enlargement operation ... any of the leaders' in the ...
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23 views

Mathematics branch concerned with availability of information

Is there a branch of mathematics that study about availability of information? For example, if I want to search for something on the internet, is there a branch of mathematics that can predict how ...
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1answer
36 views

Analysis of Kullback-Leibler divergence

Let us consider the following two probability distributions ...
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45 views

Upper bound on the entropy of a sum two random variables

Let $X$ be a random variable such that $|X| \leq A$ almost surely, for some $A > 0$. Let $Z$ be independent of $X$ such that $Z \sim {\cal N}(0, N)$. My question is: How large can the entropy ...
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1answer
19 views

mutual information of coupled variables

I have been looking for a method for evaluating the mutual information between a combination of source variables, $X_0, X_1$ and a target variable, $Y$. $$I(Y;X_0,X_1)$$ When I look on wikipedia's ...
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49 views

Ways to code two arbitrary binary strings into one without loss of information, and relevant bounds

If the title was not clear, I'm examining methods of taking two binary strings as input and outputting one binary string in such a way that the two original strings can be extracted from the output, ...
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1answer
41 views

partition with infinite entropy?

Let $P$ be an infinite partition of the interval $[0,1]$. Let $P$ have elements $I_i$ which has Lebesgue measure $m(I_i)$. Then the entropy of $P$ is defined by $\sum_i -m(I_i)\log m(I_i)$. Can this ...
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16 views

Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Let $p_0(x)$ be some arbitrary but fixed input probability distribution. The mutual information between the input ...
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46 views

Special case of Kullback-Leibler additivity

I have three random variables $X,Y,Z$. If $(X,Z)$ are an independent pair and $(Y,Z)$ are an independent pair, then the additive property of the Kullback-Leibler divergence says $K(X,Z|Y,Z) = K(X|Y) ...
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1answer
16 views

Confusion about non-negative mutual information

The formula I was given for calculating information for a specific stimulus $s_x$ is: $$I(R,s_x) = \sum_i p(r_i|s_x) \log_2{p(r_i|s_x)\over p(r_i)} $$ It was also said that information is always ...
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197 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 ...
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122 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 ...
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1answer
50 views

Upper Bound on Mutual Information

I am interested in an upper bound on mutual information that I have been encountering frequently in the statistics and probability literature. I have yet to see the "purest" form of the inequality, so ...
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34 views

Mutual Information for Gaussian Process (and also Fano's Inequality)

According to this presentation: Bounding Gaussian Process Information Gain we have a closed-form expression for the information gain as follows: $$ I\left(\vec{y} \mid f\right) = \frac{1}{2} \log\det ...
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27 views

Do 2 timeseries represent the input better than one?

I only have a very basic familiarity with signal processing and information theory so I'm sorry if this is a very straight forward question. I have a very brief input signal and two timeseries as ...
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1answer
92 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 $x$ ...
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1answer
21 views

Bounds for Mutual Information

$V_1$, $V_2$ be two binary strings with equal number of bits (say the length is $l$). Then the mutual information of $V_1$, $V_2$ can be defined as: $I(V_1;V_2)$ = $\sum_{y \in Y} \sum_{x \in X} ...
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100 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 ...
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48 views

Compressing binary numbers

If I have a arbitrarily long random binary number with the condition that the probability that a given digit is 0 and 1 is 1/4 and 3/4, respectively. What is the best way to compress this into a ...
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1answer
40 views

Calculate Huffman code length having probability?

Having an alphabet made of 1024 symbols, we know that the rarest symbol has a probability of occurrence equal to 10^(-6). Now we want to code all the symbols with Huffman Coding. How many bits ...
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1answer
42 views

Using binary entropy function to approximate log(N choose K)

I am not a mathematician and struggling with the exercises while reading this book Information Theory, Inference and Learning Algorithms. The author introduced the binary entropy function at the ...
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1answer
43 views

How is the formula of Shannon Entropy derived?

From this slide, it's said that the smallest possible number of bits per symbol is as the Shannon Entropy formula defined: I've read this post, and still not quite understand how is this formula ...
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1answer
23 views

Equality of Information Gain and Mutual Information

I am curious about definition of information gain and mutual information in the context of feature selection. If looks like two these measures define exactly the same thing, however I didn't find ...
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46 views

Shannon-Fano analysis, Binary-search-like

Prove that the codewords of the Shannon-Fano code satisfy $l_i \leq \left \lceil \log _2 \frac1{p_i}\right \rceil$. Elementary wording: given positive numbers in descending order $p_1,...,p_n$, ...
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12 views

Few questions regarding applications of conditional entopy

I have the idea of entropy and conditional probability etc and having few conditional entropy related questions: 1. What does actually conditional entropy $H(Y|X)$ mean? How can we explain this term ...
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1answer
24 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 ...
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1answer
47 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)] ...
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154 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 ...
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1answer
71 views

Is there a combinatorial explanation for this identity related to Kraft's inequality?

Kraft's inequality involves the quantity: $$\sum_{x \in X} \frac 1 {b^{\ell(x)}} \tag 1$$ Where we are considering a code mapping symbols in the alphabet $X$ to strings in an alphabet of $b$ ...
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3answers
144 views

Similarity between two probability distribution

I am not sure how to put the question. I am not even sure if this question makes sense at all. I know that the similarity of two discrete (or continuous) distributions can be quantified by ...
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1answer
34 views

A question on Markov chain

Suppose for two random variables $X$ and $Y$ we have $X\perp\!\!\!\perp Y$ and also assume that three random variables $X$, $Y$ and $Z$ form the following Markov chain: $X\to Z\to Y$. Do these two ...
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1answer
45 views

Entropy of sum is sum of entropies

Having $X$ and $Y$ discrete random variables above finite set. Z is defined as $Z=X+Y$ when does the following happen: $$H(Z)=H(X)+H(Y)$$
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38 views

Good low-rate, short-length block codes

I am highly unsure whether this question is appropriate for this site (as it is at no point a math problem), yet searching in the stackexchange universe for similar topics showed the most hits on ...
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34 views

What is the “true” entropy of a binary string?

Consider an infinite binary string $\sigma$ and define its entropy $$H_1 = -(p_0 \log_2 p_0 + p_1 \log_2 p_1)$$ with $p_i = \lim_{N\rightarrow \infty} N(i)/N$, $N(i)$ the number of $i$'s among the ...
3
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19 views

Entropy of Group Action by Knowing Finiteness of Unidimensional Subaction

I've been trying to solve the following problem " Considering a measurable dinamical system $(X, \mathcal{B}, \mu, \mathcal{T})$ where $\mathcal{T}$ is an action of a semigroup $G = N^d$ on $X$ for ...
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40 views

Problem with calculating probability of symbols

I've a $100 \times 100$ binary matrix it`s constructed with this probability table : i want to apply extended Huffman on this matrix my idea is to compress each column individually . - so starting ...
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30 views

Is there an information theory for continuous time signals?

Information theory books talk about entropy and mutual information of discrete time processes, such as a sequence of symbols sent with a symbol period $T_s$ and there received sequence. Can we talk ...
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66 views

What's the Shannon entropy of the prime numbers?

Here's a note that calculates it as 1. Do you know of any other calculations? http://www.math-math.com/2014/05/shannon-entropy-shannon-entropy-of.html
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75 views

How is Goedel's 1st incompleteness theorem related to the Axioms of a theory

i am thinking of various connections and formulations of Goedel's 1st incompleteness theorem. Apart from connections to Turing's Halting Problem and Algorithmic Complexity Theory, i am looking for ...
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59 views

Doubts in Bayes' Theorem

I meet one problem on the probability and statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$, and the probability space $(X,S,\lambda)$. And there exsit functions ...
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36 views

Mutual Information in an Binary Erasure Channel

Imagine a Binary Erasure Channel as depicted on Wikipedia. One equation describing the mutual information is following: $$\begin{array}{rcl} I(x;y) &=& H(x) - H(x|y) \\ &=& ...
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15 views

Kullback -Leibler between multiplication of probability measures

Given the probability measures $P_1, P_2, Q_1, Q_2$. prove that: $$D_{KL}(P_1 \cdot P_2 \parallel Q_1 \cdot Q_2) = D_{KL}(P_1\parallel Q_1) \cdot D_{KL}(P_2 \parallel Q_2)$$