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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Questions Regarding Mutual Information

I've been conducting a small experiment to test a few of my interpretations about mutual information, and I'm running into some difficulties. I've created some MATLAB code that basically makes two ...
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Capacity of uniform noise channel

I'm pretty sure I've made some error in reasoning in this exercise because it gets a little bit too easy. Let $X$ be independent from $N$ where $N$ is uniformly distributed over $[-1,1]$. Suppose ...
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a question about entropy of run length coding

I'm doing an exercise from chapter two of $\textit {elements of information theory}$. Here is the problem and its solution, . I'm not very clear about the equation 2.36 or say why does the equation ...
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Entropy for three random variables [duplicate]

I'm just working through some information theory and entropy, and I've come into a bit of a problem. In many texts, it's easy to find the "chain rule" for entropy in two variables, and the ...
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32 views

conditioning reduces mutual information

$I (X, Y |Z) >I (X,Y)$ can happen, for example $X$, $Y$ independent bits and $Z = X+Y$. How do you show this fact using Venn diagrams? Conditioning would mean removing the mass of conditioning ...
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26 views

Sum of Coding rates of encoders accessible by decoder must be at least “h”, in order for “r” to be admissible. How?

Let we have one source with information rate "h" and 4 encoders with information rate r1, r2, r3 and r3 and r=[r1...r4]. Following conditions are necessary for r to be admissible. How? r1+r2 >= h ...
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38 views

Required bits to communicate a partial order?

Suppose that you have a ranking (i.e. a strict complete partial order) over $n$ different objects, so that the objects can be ordered as $a>b>\cdots>n$. You want to communicate the exact ...
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24 views

the counterexample of the pinsker inequality.

if P and Q are probability measures over a set X, and P is absolutely continuous with respect to Q,then set $$ D_{\mathrm{KL}}(P\|Q) = \int_X \ln\frac{{\rm d}P}{{\rm d}Q} \, {\rm d}P, $$ set ...
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25 views

Schur complement for Square root information matrix

Consider a joint information matrix I over X and Y (both vectors). Now, I would like to get the marginal information matrix for X: $I_x$. This can be of course performed via Schur complement. Now ...
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21 views

quantization of a discrete probability source

The alphabet of a memory less source is $A=\{-5,-3, -1, 0, 1, 3, 5\}$ with corresponding probabilities $\{0.05, 0.1, 0.1, 0.15, 0.05, 0.25, 0.3\}$. If I know that the source can be quantized ...
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18 views

Best possible approximation of P(X,Y)

I want to show that $\min_Q D(P_{Y|X} || Q | P_x) = I(X;Y)$ and I've arrived at a question. Here $D$ is the KL divergence or relative entropy and $I$ is of course the mutual information. ...
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45 views

What is an intuitive explanation for Birkhoff's ergodic theorem?

If I'm not familiar with measure theory, what is a good way to understand the idea behind the definitions involved, the interpretation of the theorem, and the proofs thereof? Particularly, it's not ...
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31 views

Is there a means of calculating the entropy of a series of bits that takes correlation into account?

A common expression for calculating the entropy of a series of bits appears to be: $$-\sum_{i}{P\left (x_i\right )log_b\left (P \left (x_i\right )\right )}$$ This seems to fail (or my intuition of ...
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33 views

Conditional typicality (information theory) intuition

Reading Network Information Theory, Gamal. It's a bit terse at times and I'm trying to get an intuition for conditional typicality. The conditional typicality lemma states Let $(X,Y) \ \tilde{} \ ...
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17 views

Fisher Expected Information for a Gaussian Process model

Suppose I have a two dimensional Gaussian process model (GP), defined by a squared exponential correlation function s.t: $$R(x_{i},x_{j}) = \exp\left(-\frac{|x_{i} - x_{j}|^2}{2}\right).$$ I am ...
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15 views

Prove the identity $nH(X_1,…,X_n)=…$ for any $n \geq 2$

How can I prove that the identity $$nH(X_1,...,X_n)= \sum_{1\leq i_1 < i_2<...<i_n\leq n+1} H(X_{i_1},...,X_{i_n})+\sum_{i=1}^{n} H(X_i|X_j,j\neq i)$$ stands for any $n \geq 2$ For n=2 we ...
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sufficient and necessary condition for equality between conditional mutual information and unconditional one.

Suppose $X, Y, Z$ are three discrete random variables. Is there a good sufficient and necessary condition for $I(X;Y|Z) = I(X;Y)$? Usually the LHS can be bigger or smaller than the RHS, but if Z is ...
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26 views

Min/Max number of inequalities needed to determine the order of $n$ numbers

We are given an ordered $n$-tuple of positive real numbers $R=(r_1,..r_n)$. A $k$-inequality is an inequality of the form $x_1<x_2<...<x_k$ where $x_1,..,x_k$ are in $R$. For example, for ...
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22 views

Blackwell's informativeness criterion

Let $a=(a_1,\dots,a_i,\dots,a_n)$ be a probability vector, i.e. $\forall i: a_i\ge0$ and $\sum_i a_i=1$. Suppose $b$ is another $n$-dimensional probability vector. Is it true that there always ...
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43 views

Ratio between forward and reverse conditional probability

I have a probability distribution $p(Z | X)$ from which I can easily sample and compute the probability at every value for $Z$ and $X$. The inverse distribution $p(X | Z)$ however can be very complex ...
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How to prove H(X,Y) $\ge$ H(Z)?

I'm solving a problem from elements of information theory, 2nd. I got stuck by question(c) and actually, I've checked the answer, here it is: How to prove the inequality from the answer that is ...
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38 views

Calculating the information per symbol of a markov chain source

I have a 4-state 2nd order markov chain source with symbols 0 and 1. I have all the transition probabilities and have worked out the probabilities of each state. How do I go about finding the amount ...
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27 views

Is there any differentiable function $f$ that approximates the “entropy” of a set of numbers $S$?

Where entropy is some measure of the degree of randomness/disorder in a given set of numbers: $S = \{a_1, a_2, ..., a_i\}$ For example, the set $S_{high} = \{4,0,2,5,8,3,7,2,5\}$ has a high degree of ...
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114 views

Coupon Collector Problem for Non-Uniform Coupons: On the number of missed Coupon

Suppose $\mathcal B=\{1,2,\ldots,b\}$ is the set of all possible coupons, with $\mathbf p = ( p_1,p_2,\ldots,p_b)$ assigning the probability of occurrence for all coupons in $\mathcal B$. The ...
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What is the variance of self-information (or surprisal)?

The self-information of an outcome $x_i$, or surprisal, is defined as: $$ I(x_i)=-\log P(x_i), $$ where $P$ means probability. This way, the Shannon entropy can be seen as the "average" or "expected" ...
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Statics maximization

I really need a help to compute the following maximization problem. $$\max _{p(x), E(x) \leq \alpha} \int_x \int_y \frac{p(x,y)^2}{p(x)p(y)} dx \,dy$$ Suppose that : $$p(y|x)=\frac{1}{\sqrt{2 \pi ...
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36 views

data processing inequality using non-deterministic functions

Generally data processing inequality says that the entropy cannot increase on applying a function f, or to be precise $H(f(X))\leq H(X)$ (also it is reversed if we know the function is k-to-1 so there ...
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43 views

How a Galois Field is used to construct a Hamming Parity Matrix

I'm trying to understand how Matlab is generating their Hamming parity matrix. The default according to the documentation is GF(2^m), where m=3. Hamming(7, 4) parity matrix from Matlab ...
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35 views

Largest positive eigenvalue of a matrix

I am dealing with the Capacity of constrained noiseless communication channels. It has been said that the channel capacity of such a channel is $\log{\lambda}$, which $\lambda$ is the largest positive ...
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Constructor theory distinguishability

In David Deustch and Chiara Marletto's Constructor Theory of Information (section 5) a set of attributes $S$ is defined as distinguishable if the task of transforming each attribute $x$ of $S$ into an ...
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Structure of equientropic transformations

Given a probability vector $v=(v_1,\ldots,v_n)$ with $1\geq v_i\geq 0$ and $\sum_{i=1}^n v_i=1$ its entropy can be defined as: $$ H(v):=-\sum_{i=1}^nv_i\log v_i $$ I wonder what is known about ...
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32 views

Identifying a Markov chain

This is a very basic question in the theory of Markov chains and I'm just not sure how to prove it mathematically. Say we have random variables $X, Y$ that are correlated and we have a possibly ...
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How to calculate the mutual information between two outputs of Rayleigh fading channels

We have the two channels: $$X_{a,i} = H_{i}s_{i} +N_{a,i} \\ X_{b,i} = H_{i}s_{i} +N_{b,i} $$ for $1 \leq i \leq n$, where $H_i$ denotes the i.i.d. channel coefficient and is a zero-mean complex ...
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Relative Entropy decomposition reference

may I ask for some reference pointers? My bad as I got a classic case of losing my reference and thus unsure what I wrote was right or wrong. I tried looking my old references and internet and didn't ...
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42 views

Proving certain aspects of Entropy

I am trying to prove three properties of entropy. $1)$ $H(X|Y,Z)\le H(X|Y)$ $2)$ $H(X|Y,Z)\le H(X,Y)$ $3)$ $H(X,Y,Z)+H(Y)\le H(X,Y)+H(Y,Z)$ I have proved the third one, but it is based on part 1. ...
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47 views

Simple information theory question: where is this equation coming from?

I am reading a simple example of a joint distribution that looks like this: ...
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35 views

Comparing two mutual information expressions

Given the following Data Processing Inequality $$X\rightarrow Y\rightarrow Z$$ one can say that $$I(X;Y) \geq I(X;Y|Z)$$ Intuition tells me this is not correct since conditioning reduces entropy and ...
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43 views

Markov chain and mutual information

If $X\rightarrow Y\rightarrow Z$ follow a Markov chain, then we have the following properties$$I(X;Z)\leq I(X;Y)$$ where $I$ is the mutual information expression. Intuitvely I agree. I want to ...
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29 views

Entropy is upper bounded by cardinality of the random variable

How can one prove that the entropy of random variable $X$ is upper bounded by $\log|X|$? I tried the following $$H(x) = - \sum_x p(x)\log p(x)$$ $$ \leq - \sum_x p(x)\sum_x\log p(x)$$ $$= - ...
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Kullback-Leiber divergence for two simple probability vectors

For any probability vectors $ p= (p_1,...,p_K) $ and $ q=(q_1,...,q_K) $ representing monotonically increasing functions $ x-1 $ and $ ln(x) $ respectively what is the KL divergence? $$ KL(p||q) = ...
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Upper bound for conditional entropy? (for information theorists)

Assume we have $X$ and $Y$, zero mean and independent from one another. Assume that the variance $$\text{Var}(X)= P$$ $$\text{Var}(Y)= Z$$ I need to show that $$h(X|X+Y) \leq \frac{1}{2}\log ...
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Have Information Theoretic results been used in other branches of mathematics?

consider this a soft-question. Information Theory is fairly young branch of mathematics (60 years). I am interested in question, whether there have been any information theoretic results that had ...
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Explanation of Information double summation within Normalized Mutual Information

The Normalized Mutual Information NMI calculation is described in deflation-PIC paper with the applicable formula copied to the screenshot shown below. My question is specifically about the double ...
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33 views

Binomial coefficients bounded by entropy exponential

So I'm trying to prove that for $\frac{1}{2}< x \leq 1$ we have $$\sum_{\lceil nx \rceil}^{n}{n \choose k} \leq 2^{nh(x)}$$ I've managed to prove that $$\sum_{0}^{\lfloor nx \rfloor}{ n\choose ...
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Capacity of Z Channel

Calculating the capacity of the Z channel (binary asymmetric channel) here, the entropy $ H(Y)$ isn't supposed to be $ H(Y)=H(a+(1-a)p,(1-a)(1-p))$ ? What's the reason for having $H(Y)=H((1-a)(1-p))$ ...
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Metric Entropy Upper Bounds

In the paper Information-Theoretic Determination of Minimax Rates of Convergence the authors present Theorem 3 as follows: If $M_2(\epsilon)$ is the $\ell_2$ packing entropy of a density class ...
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Convexity of mutual information $I(X;Y)$ in conditional $p(y \mid x)$

I'm trying to understand the proof that $I(X;Y)$ is convex in conditional distribution $p(y \mid x)$ - from Elements of Information Theory by Cover & Thomas, theorem 2.7.4. In the proof we fix ...
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Question regarding the Entropy of a probability mass function

I assume that the entropy, $E$, of a probability mass function (pmf), $p(X)$, of a discrete random variable, $X$, is computed as: $$\begin{align}\mathbb{E}(p(X)) &= -p(X = x_1)\log[p(X = ...
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50 views

Relation between Genetic Algorithm and Information theory

Can anyone suggest me some references (papers, books, lecture notes) on the relation between GA and Information theory?
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33 views

Convexity of $I(X;Y)$: why $H(Y)$ convex in $p(y)$ $\Rightarrow$ $H(Y)$ convex in $p(x)$

I would like to understand the proof that mutual information $I(X;Y)$ is concave in $p(x)$ - as presented in Elements of Information Theory by Cover & Thomas, theorem 2.7.4. Here's the proof from ...