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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Question about the Shannon Entropy formula

I have a basic question about the Shannon Entropy formula. In fact it's so dumb that I didn't dare ask it in the class because I don't understand the text books. Here's the formula: $$H(X) = -\...
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19 views

Constraints on Mutual Information Independence Test

Suppose all variables are binary for the sake of simplicity. There is a theorem about mutual information (MI) and a distribution $\chi^2$. Given a data set D with N data points, if the hypothesis ...
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20 views

p(a,c) vs p(a∧c)

In this paper: https://www.aclweb.org/anthology/J/J16/J16-2006.pdf, the author breaks the Pointwise Mutual Information of a phrase up into several components: They use the ...
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16 views

Decomposition of Shannon conditional mutual information as seen on Wikipedia [Solved]

I am looking for a proof for a formula seen on Wikipedia: \begin{align} I(X;Y|Z) & = H(Z|X) + H(X) + H(Z|Y) + H(Y) - H(Z|X,Y) - H(X,Y) - H(Z) \\ {} & = I(X;Y) + H(Z|X) + H(Z|Y) - H(Z|X,Y) - H(...
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55 views

Difference of Entropy of two-dimensional Gaussians

I encountered a putative contradiction. Assume we have two 2-dim. Gaussian variables $z_1 = (x_1, y_1)$ and $z_2 = (x_2, y_2)$ with all components being independent, normal distributed variables: $x_1,...
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Representing pairwise-independent but not independent occurrences with venn diagram

For $A,B$ and $ C $ partially pairwise independent occurrences (i.e. $I(A;B)=0$, $I(A;C)=0$ ), it is not true to say that $I(A;B,C)=0$, since $I(A;B,C)=I(A;B)+I(A;B|C)$ [<-this is not correct, see ...
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Proof and physical meaning of $I(X;Y) \leq \min \{ \log| \mathcal X |, \log| \mathcal Y | \}$

The inequality holds, $$I(X;Y) \leq \min \{ \log| \mathcal X |, \log| \mathcal Y | \}$$ where $I(X;Y)$ is the mutual information. I know that $H(X) \leq \log| \mathcal X| $ is an upper bound on ...
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37 views

Decomposition of Random Variable (Information)?

I am wondering whether the following idea or something similar appears in a field such as statistics or information theory(?). Take a random variable $Y$ which takes value $1$ or $2$ with equal ...
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21 views

Information Channel Capacity

1) Suppose a Noiseless Binary Channel, who's input is reproduced exactly at the output. Let X be the transmitter and Y the receiver (i.e (X=0----->Y=0 and X=1-----> Y= 1)) I understand intuitively ...
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66 views

Help me understand the proof for Shannon's Theorem 4 (regarding number of sequences of various probabilities) in the original paper

I'm reading Shannon's 1948 paper where I encountered Theorem 4: $$ \lim \limits_{N \to \infty} \frac {\log n(q)} N = H $$ In Appendix 3 after proving Theorem 3, Shannon proves Theorem 4 by saying: ...
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19 views

Quantizer Functions

Let $Y \sim P_Y$ with variance $P^{\alpha_1}$ $P>1$. Assume $n \sim P_n$ with variance $P^{\alpha_2}$ for any $\alpha_2 \le \alpha_1$. Let $\mathcal{Y}$ be the set over which the random variables $...
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34 views

mutual information and combinatorics

\begin{align} &\mathrm{H}\left(\frac{1}{2^{k}}\right) \\[3mm]&\ \!\!\!\!\!\!\!\!\!\! - {1 \over 2^{k}}\left\{% {k \choose 0}\mathrm{H}\left(\left[1 - \epsilon\right]^{\,k}\right) + {k \choose ...
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35 views

Common Estimates Suggestions

Consider two Markov Chains $X-Y_1-Z_1$ and $X-Y_2-Z_2$ defined on same alphabet space $\mathcal{X}$, such that $Z_1= g_1(Y_1)$ and $Z_2=g_2(Y_2)$ for some functions $g_1,g_2$. Assume further that ...
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77 views

Splitting a file into $m$ pieces of size $1/n$, such that any $n$ pieces allow you to recover the file?

Let's say we have a file (which we could define as a finite sequence of 0's and 1's (or any other two symbols)). For $m > n$, can you create $m$ pieces (which are themselves files), each $\frac 1n$...
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40 views

Minimum distance for large codes

The minimum distance is easy to compute, and so determine the error correcting/detection capabilities of a code, by enumerating all possible pairs of codewords and computing the hamming distance ...
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KL Divergence between the sums of random variables.

The relative entropy or Kullback–Leibler distance between two probability density functions $g(x)$ and $f(x)$ is defined as $$D(g\|f) = \int_{x} g(x)\log\frac{g(x)}{f(x)} dx .$$ We have two random ...
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40 views

Explaining something in Kraft-McMilan inequality proof

I was asked to present the Kraft-McMillan inequality, but I have trouble in understanding why in the following segment of the proof: there is a $$k\ge0 $$such that: $$ a^{-l_1}+a^{-l_2}+.....+(k+1)a^{...
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29 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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32 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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41 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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2answers
183 views

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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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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21 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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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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45 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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23 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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126 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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81 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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47 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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33 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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50 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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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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36 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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23 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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21 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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416 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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23 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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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
25 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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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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23 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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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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71 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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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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20 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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28 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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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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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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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 ...