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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2
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11 views

Help me construct a bijection $g: \left(1..n\right)^m \to \left(1..n\right)^m$, with influence of each component of $x$ in each component of $g(x)$

Let $x \in \left(0..n-1\right)^m$. I want to construct a bijection $g$ : $\left(0..n-1\right)^m \to \left(0..n-1\right)^m$ such that if we know $m'$ components of $g(x)$ and $m-m'$ components of $x$, ...
0
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1answer
13 views

Mutual Information of Coupled channels

I am trying to determine the what the expression for the mutual information of following system would be; $Y_1$ = ($X_1$ + $\eta_1$) + A($X_2$ + $\eta_2$) $Y_2$ = B($X_1$ + $\eta_1$) + ($X_2$ + ...
0
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0answers
28 views

Is there an alternative encoding scheme to binary where similarity of pattern correlates with size of number?

If I compare binary for 7 111 and binary for 8 1000 there is no correlation between these two patterns that suggests that ...
0
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1answer
15 views

Proof of an inequality with entropy and mutual information.

Entropy of a random variable (a) is (h) : $H(a) = h$. Mutual information of (a) and (b) is (3h/4) : $I(a;b) = 3h/4$. Mutual information of (a) and (c) is (3h/4) : $I(a;c) = 3h/4$. It is needed to ...
0
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1answer
31 views

Linearity of codes

Assuming $C$ is a binary linear code and let $a$ $\notin $ $C$ be any vector. Show that $C$ $\cup (a + C) $ is also linear. I know that for any $C_1,C_2 \in C $ then $\alpha C_1 + \beta C_2 \in C$ ...
1
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0answers
44 views

Vector distance of binary

Suppose $\overline{u},\overline{v},\overline{w},\overline{x}$ are four binary vectors, pairwise distance d apart. Show that d must be even, there's exactly one vector which is a distance $d\over 2$ ...
0
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0answers
21 views

What are some techniques of specifying a molecules structure using the least amount of information?

For instance say I have a water molecule I can describe it's structure by two bond lengths and a bond angle. Are there any neat math tricks or representations of objects that I could use to describe ...
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20 views

Classification of pairs of random variables up to asymptotic equivalence

I will begin by explaining how entropy classifies random variables up to "asymptotic equivalence". Then I will extend the notion of asymptotic equivalence to pairs of random variables. Information ...
1
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23 views

Constructor Theory: Why does $y\subset z$ imply $y \not \bot z$?

In the paper Constructor Theory of Information, page 11. We are told that a set $X$ of attributes are distinguishable if the following is a possible task: $$\{ x\rightarrow \Psi_x | x\in X \}$$ where ...
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2answers
25 views

Mutual information expressed as Kullback-Leibler divergence

My lecturer defines the mutual information: $$ I(X;Y\mid Z) = D_{KL}\big(p(X,Y\mid Z)\parallel p(X\mid Z)\;p(Y\mid Z)\big)$$ Is this correct? I feel like it doesn't really make sense to say that; ...
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12 views

Proof that the inequality with mutual information and conditional mutual information is not true always.

It is needed to prove that inequality $I(a:b) \le I(a:b|x) + I(a:b|y) + I(x:y)$ (where I(a:b) is mutual information and I(a:b|x) is conditional mutual information) is true not for all sets of random ...
2
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0answers
67 views
+50

Using Markov Chains to simplify mutual information expressions

I read a paper in Information Theory which claims that the following sum of three mutual information expressions $$ I(Y_1;X_1,X_2,X_3)+I(Y_2;X_2,X_3\mid Y_1,X_1)+I(Y_3;X_3\mid Y_1,Y_2,X_1,X_2)\tag1$$ ...
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14 views

Task about Hartley information(logarithm from cardinality of the set) of a number of paths in a graph and about limit linked with this information.

Let $L_n$ be the number of all paths of length n in a directed graph(below). It is needed to find $lim_{n \to \infty}\chi(L_n)/n$ where $\chi(L_n)$ is Hartley information in $L_n$ set. (If I am not ...
0
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1answer
20 views

Existing of a distribution of three random variables that have conditional mutual information with defined properties.

I have two similar questions: 1)Does exist a distribution of three random variables such that: $I(a:b) = 0$ and $I(a:b|c)>0$ (where $I(a:b)$ is a mutual information and $I(a:b|c)$ is a ...
5
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0answers
42 views

Guessing number in set 1-100 with weighted questions.

It is needed to guess number from 1 to 100. I can ask questions and get answers:"yes" or "no". For the "yes"-answer I must pay one dollar, for the "no"-answer - two dollars. How many dollars should I ...
1
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1answer
21 views

Why does the information content of a less probable event yield more bits than a more probable one?

I gather for a series of four coin flips, if we get $H,H,H,H$, this has a probability of $\tfrac{1}{16}$, so we have information content $$\log_2 \frac{1}{\frac{1}{16}}$$ But for the rest of the ...
1
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0answers
40 views

Proof of an inequality about set in four-dimensional space

It is needed to prove that for set $A \subset N^4$ $3\log|A|\le \log|\pi_{1,2,3}(A)|+\log|\pi_{1,2,4}(A)|+\log|\pi_{1,3,4}(A)|+\log|\pi_{2,3,4}(A)|$ where $\pi_{i,j,k}(A)$ is a projection of $A$ on ...
0
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0answers
26 views

The mutual information rate spectrum

Definition: $\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...
0
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1answer
24 views

Decomposability of Inf. Entropy applied to corpus statistics

Can you help me understand the additivity property of IE? I think I may have found a violation but it may be that I'm just misunderstanding the maths. I'm attempting to achieve a meaningful ...
0
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1answer
37 views

How to understand the product of two conditional probabilities?

I am struggling a little bit with making sense of the distribution of bigrams in an artificial language I randomly generated from english. Every word occurs with an equal probability but the ...
1
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1answer
22 views

Confused By Information Diagram

I'm working on information theory and I get confused by the information(venn) diagram. Is there any one help me with the following confusion? As defined, mutual information $I(X;Y)$(Figure a) is ...
1
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1answer
19 views

Understanding Conditional Mutual Information

Recently I'm working on conditional mutual information and I'm trying to prove the following property: I(X;Y|Z,W)<=I(X;Y|Z) The property seems obvious to me: ...
0
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1answer
19 views

Finding error control capability of Hamming distance

I have known how to calculate the Hamming distance between two message codes. But I don't know how to get the error control capability. In one case I have hamming distance of: $$ d = 8 $$ Errors ...
0
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1answer
20 views

Mutual Information Gaussian

Suppose $x$ has a multivariate Gaussian distribution given by $\mathcal{N}(\mu , \Sigma)$. How do you express the mutual information between two coordinates that is $\mathrm{I}(x_i;x_j)$ , as a ...
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0answers
11 views

Proof of inequality about mutual information

There is some joint distribution of random variables $(a, b, c, x, y)$. It is known that $I(a;b|c) = I(a;c|b) = I(b;c|a) = 0$ It is needed to prove this inequality: $I(a;b) \leq I(a;b|x) + I(a;b|y) + ...
2
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0answers
17 views

Kolmogorov complexity inequality

Prove, that KP (x) ≤ KS (x) + log KS(x) + 2 log log KS (x) + O(1). Please tell me in which direction to think.
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0answers
11 views

Optimal Kolmogorov complexity

Let computable function U is the best way to describe to Kolmogorov complexity. Prove that the mapping V, determined crucial for any word p as V (p) = U (U (p)), is also optimal way to describe the. ...
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22 views

Proof of a fact about mutual information and entropy

It is needed to prove that for distribution (a, b, c) such that $I(a;b|c) = I(a;c|b) = I(b;c|a) = 0$ exists a random variable $d$ such that $H(d) = I(a, b, c)$ and $a,b,c$ are independent with ...
1
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1answer
29 views

Proof of inequality with entropy

I can not prove this inequality $2H(a,b,c) \leq H(a,b) + H(a,c) + H(b,c|a),\ H-entropy $. I tried do it by using chain rule and this inequality $H(X|Y) \leq H(X;Y)$ but without any success. Please ...
0
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1answer
21 views

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 ...
0
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0answers
20 views

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 ...
0
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1answer
30 views

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 ...
1
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2answers
32 views

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 ...
1
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1answer
25 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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0answers
24 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 ...
0
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2answers
35 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 ...
0
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1answer
22 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 ...
0
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0answers
21 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 ...
0
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0answers
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 ...
1
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1answer
17 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. ...
3
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1answer
43 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 ...
0
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3answers
30 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 ...
0
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1answer
27 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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0answers
16 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 ...
0
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0answers
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 ...
2
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0answers
26 views

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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1answer
25 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 ...
2
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1answer
16 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 ...
0
votes
1answer
39 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 ...
2
votes
2answers
66 views

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 ...