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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Formula for proportion of entropy

Let's say we have a probability distribution having 20 distinct outcomes. Then for that distribution the entropy is calculated is $2.5$ while the maximal possible entropy here is then of course $-\ln(\...
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16 views

How to use the log-sum inequality to prove convexity of KL-divergence?

I'm trying to read up on information theory, and found the following: http://homes.cs.washington.edu/~anuprao/pubs/CSE533Autumn2010/lecture3.pdf Which states that the convexity of KL-divergence can ...
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28 views

Is there a name for the property of a code where symbol “space” is left unused?

For example, say I have the symbols A, B, C and D. If I encode these as A = 1, B = 01, C = 001 and D = 0001 (for a very simple example), I have a very simple prefix code. However, I know straight ...
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1answer
35 views

Help in proving inequalities in information theory-Kraft-McMillan

I was given a task of proving some inequalities that are related to Kraft-McMillan's inequalities, and i have been scratching my head for quite some time trying to prove it: $$ F(x)= \frac{1}{1-Q(x) }...
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1answer
20 views

Deriving the Power Spectral Density of a Maximum Entropy Process

In Elements of Information Theory, Chapter 12, Section 6 Burg's Theorem is derived: Presented with the first $p$ values of the autocovariance function $R(k) = E[X_i X_{i+k}]$ a stochastic process ...
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15 views

Cumulants of square of Poisson distribution

I'm writing up a derivation of an expression for mutual information between weakly interacting Poisson processes. I'm running into an expression that looks like this: $$\log\mathbb{E}\left[e^{\...
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16 views

How can I show that any integrable Passband or Baseband signal is also a finite energy signal?

I have supposed that, as the definition of a baseband/passband signal says, the function x(t) is integrable, continuous and bounded due to the fact that it forms a Fourier Transform pair with x'(f) (...
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89 views

I need help in Entropy question (information Theory)

I need help in proving an inequality: $1 - H(p,q) \leq |p-q|$ and $H(p,q) \geq 2p$ and $H(p,q)\geq 2 \min(p,q)$? Thanks in advanced!
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1answer
41 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
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1answer
19 views

MLE in introductory probabilistic information theory

Consider sending a bit that is either $\{0,1\}$ through a noisy symmetric channel, such that for a given input $x$ and a given (potentially noisy) output $y$, $\forall i,j \in \{0,1\}. P(y = i | x = j)...
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70 views

Campbell's Source coding

In the usual Shannon's source coding problem one chooses code words that minimize $E[L]:=\sum_i p_il_i$ over all $L=(l_1,l_2, \dots), l_i\ge 0$ such that $\sum_i e^{-l_i}\le 1$ (Kraft inequality), ...
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1answer
24 views

Significance of Convex Sets for I-Projection

I have been reviewing the literature on information theoretic methods in statistics, and in particular, the method of I-projections. Given a discrete, finite alphabet $\mathcal{X}$, let $\prod$ denote ...
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1answer
44 views

why can't sort 12 elements in 29 comparisons

The information theoretic lower bound for sorting 12 elements is using 29 comparisons, but actually we can't sort them in less than 30 comparisons. My problem is that why we can't reach the ...
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2answers
99 views

Is conditional entropy ever taken to be a random variable?

In probability theory, the conditional expectation $E(X|Y)$ and variance $V(X|Y)$ er usually taken to be random variables, st. the value of $E(X|Y)$ depends on what value $Y$ ends up taking. I've ...
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0answers
116 views

How to keep up when converting between bases?

Here is a schematized binary channel that neatly conveys a decimal number. $ \require{begingroup}\begingroup \def\T {{ \cal T }} \def \Ti {{ \T \raise5mu{ \text- \scriptsize 1 } }} \def\Bx #1{{ ~ ...
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1answer
65 views

Concavity of Shanon's information

It is known fact for random variables $(X,Y) \sim p(x,y)=p(x)p(y|x)$ the mutual information is concave function of $p(x)$ for fixed $p(y|x)$. I have two confusions in interpreting the above fact: 1) ...
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3answers
79 views

Greater/lesser search with one false answer allowed

It is well known that you can determine the values of $n\geq 2$ bits using $k$ yes/no questions about the bits (for example, "is $x_1 \oplus x_3 = 1$?), even if one (but not more) of the answers ...
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1answer
30 views

Does the following function define a distance metric?

For real numeric vectors of length $N$, let $a_n \succ b_n$ be one if true and zero if false. The distance between $A$ and $B$ is $$\sum_1^N a_n \succ b_n$$ Note that this is very similar to the ...
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2answers
47 views

Proof of Cyclic Redundancy Check validity

I'm looking to understand the use of a Cyclic Redundancy Check, in combination with the mathematics behind it. So far I have 1) For any message $$M(x)\cdot x^n = Q(x)G(x) + R(x)$$ Where $Q(x)$ is ...
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4answers
110 views

Guess the number despite false answer

This is the Guess-The-Number game with a twist! Variant 1 Take any positive integer $n$. The game-master chooses an $n$-bit integer $x$. The player makes queries one by one, each of the ...
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1answer
541 views

How come that HSL can contain more information than RGB?

I have noticed weird thing when working with HSL - unlike RGB, it has some blind spots where certain value just does not matter. I'm sure we were taught about this when I had Linear algebra lectures ...
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1answer
33 views

Conceptual Question : Relationship between entropy and a technique for source coding

I want to encode the messages to a sequence of 1s and 0s (subsequently called "bits"). This is called "source coding". Shannon's source coding theory states that the entropy of a source that emits a ...
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14 views

Specific examples of Side Information?

I'm starting to apply information theory to gambling. There is something called Side information (see details in [1]), which I understand is additional information about the outs of the game. It could ...
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1answer
20 views

Proving some inequalities related to Information Theory

I've been working on some inequalities related to the information theory section of my decision theory course, and I could use some help on some of the derivations for one of the inequalities. As a ...
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18 views

Kullback-Leibner divergence true distribution

I have an image with an object which I treat as 2-dimensional Gaussian random vector with mean equal to the center of the object surrounded by, roughly, 3-sigma ellipsoid. On the other hand I feed the ...
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3answers
397 views

Do Gödel numbers have a practical use?

Is there any example of Gödel numbers being actually used in practice? If so for what purpose?
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803 views

what is the mutual information of three variables?

mutual information of tow variables is $\displaystyle\sum\sum p(x,y)\ln\frac{p(x,y)}{p(x)p(y)}$ what is the mutual information of three variables? is it $\displaystyle\sum\sum\sum p(x,y,z)\ln\frac{p(...
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1answer
15 views

Lower bound for limit of length of codeword

Here is the question I'm trying to solve. I don't really have any idea how to approach it/what theorem to use. For $ p, \lambda >0$, let $m(n,p,\lambda)$ be defined to be the least $m$ ...
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1answer
22 views

For convex $f$, why is $(p,q) \mapsto q \, f(p/q)$ convex on $\mathbb{R}_+^2$?

This fact was stated in the Wikipedia article on $f$-divergences to explain why they are jointly convex.
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1answer
68 views

$λ=log(2)$ for the tent map – which basis for the logarithm?

If $\lambda$ is the largest positive Lyapunov exponent of a piecewise linear dynamical chaotic discrete in time map, then is there a relationship between the entropy $h$ and its $\lambda$. According ...
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1answer
17 views

How to prove $2d_H(\{XY\},\{X\}\{Y\})^2 \le I(X,Y)$?

Let $X$ and $Y$ be discrete random variables. Denote the joint distribution of $X$ and $Y$ by $\{XY\}$ and their marginal distributions by $\{X\}$ and $\{Y\}$. Let $\{X\}\{Y\}$ denote the product of ...
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1answer
25 views

Log base change problem, Multivariate Gaussian differential entropy proof

I am working through a proof in this document http://ee.tamu.edu/~georghiades/courses/ftp647/Chapter7.pdf for Theorem 3 (The entropy of a multivariate Gaussian distribution): Let X = (X1, X2, · · ·...
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3answers
952 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 ...
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20 views

Kullback-Leibler Divergence (KL) and Approximation Symmetry Property

The Kullback-Leibler Divergence doesn't satisfy the symmetric property. But, it can be approximated (bounded) to such a value. in this paper: Compressing Interactive Communication under product ...
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10 views

Checking if a code can be unambiguously decoded

The source of information is A = {a, b, c, d}. More info is given in the table below. I have to find the average length of the codes, compare it to the entropy of ...
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3answers
3k views

How is logistic loss and cross-entropy related?

I found that Kullback-Leibler loss, log-loss or cross-entropy is the same loss function. Is the logistic-loss function used in logistic regression equivalent to the cross-entropy function? If yes, can ...
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1answer
19 views

The relationship between AEP and compression

So i've been reading up on AEP, and trying to get a grasp on it (and to figure out why it is important). I understand the general definitions, and that the whole idea is the knowlegde of typical ...
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1answer
41 views

Rate distortion function with infinite distortion

I am working through the problems in Elements of Information Theory by Cover and Thomas and have come across the following problem I couldn't answer. The problem is to find the rate distortion ...
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25 views

Lower bound on binomial tail

In something I am reading, the following statement is mentioned in passing as something obvious: if $X_1,\ldots,X_n$ are i.i.d. Bernoulli with parameter $1/2 + \delta$, then $\mathbb{P}(\sum_{i=1}^n ...
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29 views

Information in Filtrations

Is the “information” kept track of by filtrations the same as information-theoretic “information”? If not, is there some way the two concepts can be reconciled?
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1answer
19 views

Relative entropy between discrete and continuous random variables

Is this possible to define relative entropy between discrete and continuous random variables? Say $P$ is a discrete pmf and $Q$ is a continuous pdf, what is $D(P||Q)$?
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3answers
38 views

Approximation of an indefinite integral

Consider this integral $$\frac{1}{2d}\int_{-d}^{d}f(x-t) \, \mathrm{d}t$$ When $d$ goes to zero, $$\lim _{d\to 0} \frac{1}{2d}\int_{-d}^{d}f(x-t) \, \mathrm{d}t = f(x)$$ but what is the second ...
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8 views

Monte-Carlo estimation of Mutual Information over AWGN channel

I'm trying to solve a problem I was tasked with. Basically I have to generate a 100k 16QAM inputs and transmit them over a AWGN channel. With this I have to use the Monte-Carlo estimation to figure ...
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1answer
45 views

Proof regarding size and dimension of linear codes

The problem is stated as follows: Let C be a binary linear code of length n, dimension k and distance d and assume that C contains at least one element of odd weight. Let C' be the subset of C ...
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1answer
115 views

How much life does it take to stack your deck? (Sorting problem)

There is a card in Magic the Gathering called Lim-Dul's Vault. While it is slightly more complicated than presented, the question I would like to consider is this: Pay 1 life. Look at the top 5 ...
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1answer
116 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 ...
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1answer
34 views

Maximizing sum of logarithms (Z-channel capacity)

In the context of information theory, I am trying to maximize the following function (mutual information of the Z-channel's input and output) with respect to $p$ in order to derive Z-channel's ...