This tag is for [Entropy](http://en.wikipedia.org/wiki/Entropy_(information_theory)) in Mathematics.

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Entropy of a Markov chain (right result?)

Consider the Markov chain with state space $E=\left\{0,1,2,3,4,5,6\right\}$ and transition matrix $$ \begin{pmatrix}1/5 & 3/5 & 0 & 0 & 1/5 & 0 & 0\\0 & 0 & ...
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I am trying to solve this question but i am stuck in calculation of mutual entropy of both.please help [on hold]

Consider the outcome of a fair dice throw and let D be the number of dots on the top face. The random variables X, Y, and Z are defined over this sample space as follows: Xǫ{1,2,3,4,5,6} with {X =i} = ...
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Is there a symmetric alternative to Kullback-Leibler divergence?

I have two samples of probability distributions that I would like to compare. I have previously heard about the Kullback-Leibler divergence, but reading up on this it seems like its non-symmetricity ...
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1answer
30 views

Given a Markov chain $X \rightarrow Y \rightarrow Z$, why is $I(X;Y|Z) \leq I(X;Y)$?

A Markov chain $X \rightarrow Y \rightarrow Z$ is given, where $X,Y,Z$ are random variables characterized by the probability distribution $p(x,y,z) = p(x)p(y|x)p(z|y)$. It follows that $I(X;Y) \geq ...
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1answer
37 views

What is the exact meaning of $I(X;Y|Z)$ in Information Theory?

I am wondering: is the notation $I(X;Y|Z)$ used to denote the mutual information between probabilities of $X$ and $Y|Z$ or between $X|Z$ and $Y|Z$?
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1answer
22 views

Entropy determination regarding lossless data compression

Suppose I had many computer data files to compress losslessly and wanted to know what is the theoretical limit to each one as far as minimum filesize possible. How would a math person go about doing ...
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Entropy of Sum vs Difference of Random Variable

I am looking for a proof of the following fact Let X and X' be i.i.d on {0,1,2}(not necessarily uniform). Prove that $H(X + X' mod\;3) \leq H(X - X' mod\;3)$ where $H()$ is the standard Shannon ...
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If $g$ is a function of the random variable $X$, is it true that $H(X) = H(X) + H(g(X)\mid X)$?

I think my homework about entropy is formulated incorrectly. The question is the following: let $X$ be a discrete random variable. Show that the entropy of a function $g$ of $X$ is less than or ...
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35 views

Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...
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Entropy of convolution of measures

Let $G$ be a countable, discrete group, and let $\mu_1,\mu_2$ be probability measures on the group $G$. We define the entropy of $\mu_i$ as $H(\mu_i)=\sum\limits_{g \in G}-\mu_i(g)\log(\mu_i(g))$ ...
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1answer
35 views

Renyi entropy (zeroth order)

I am reading a book on information theory, therein has been introduced Renyi entropy of order $\alpha$ as $S_{\alpha} = \frac{1}{1-\alpha}\log(Tr\rho^{\alpha})$, where $\rho$ is density matrix. It ...
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28 views

Relative Entropy and variation formula for $C_c$

Let $R(\mu \mid \nu ) = \int_{\mathbb R} \log \frac{d\mu}{d\nu} d\nu$ for $\mu, \nu$ probability measures over $\mathbb R$. By the varational representation formula of Donsker and Varadhan we know ...
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1answer
51 views

Proving that $S=\bigcup_{j=0}^{2^k-1} S_{n-1+k}$ is a spanning set for the $2$-D Baker map

A set $S \subset X$ is a $(n,\epsilon)$-spanning set if $\forall x \in X$, $\exists y \in S $ such that $d_n(x,y)<\epsilon$. This is where we define $d_n(x,y)$ by $d_n(x,y)=\max_{0\leq k < ...
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Decomposing factorized entropy

I am trying to figure out how the equation for factorized entropy below is derived. The equation for entropy is $H(Q) = -\sum_x Q(x)\log Q(x)$ where $Q$ is a probability distribution. Let $Q(x) = ...
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Statistical Inference, Differential Geometry and Entropy

Context: Statistical Inference and Differential Geometry Let's consider a generic $ p(x;\theta) $ distribution with $ \theta $ Parameters Vector, it is obvious that $$ \int p(x; \theta) dx = 1 $$ ...
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1answer
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Conditional Entropy in rolling a dice

A 6-sided die is tossed once. Two events X and Y are defined. X is the event in which an even number comes up and Y is the event in which the number is a multiple of 3. The value of H(X|Y) needs to be ...
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1answer
28 views

Channel Capacity of a Cycle Graph

I have the following problem: Given a discrete memoryless channel $Y = X + Z \mod5$, where $X$ is selected from one of 5 symbols (0, 1, 2, 3, 4), $Z$ randomly selected from (-1, 0, 1), and $X$ and $Z$ ...
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1answer
23 views

prove a theorem about an upper bound of entropy of a random vector

There is a theorem that: if Z is any zero-mean, complex random vector with covariance $E[ZZ^H]=R_z$, then $H(Z)\leq \log|{\pi eR_z}|$, with equality holding if and only if Z has a circularly ...
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Entropy of bit position in a bit stream

8 bit strings are sent over a channel. First two bits are always 1. Last six bits can be either 0 or 1. Receiver randomly selects bit-position and reveals bit but not its position. If X is the random ...
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Proof of recursivity of Shannon's Entropy

Does anybody know a book where the proof of recursivity property of Shannon's Entropy can be found? I mean this: $$H(q_1,...,q_n)=H(q_1 + q_2, q_3,...,q_n) + (q_1 +q_2)H( \frac{q_1}{q_1+q_2} , ...
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1answer
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Calculate the Entropy Change if 5 Previously Tossed Dice Are Turned to All “1”

Relevant Equations: S = Boltzmann*ln(W) where S is entropy and W is the number of microstates. I have thought about this two ways. 1 way. Look at each die separately. Let macrostate 1 = number of ...
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1answer
105 views

Prove there exist a $p$ so that the inequality holds

I am stuck with the following problem. Given the Gaussian mixture distribution $f(\cdot)$ $$ f(x) = ...
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55 views

How to compute the topological entropy of a permutation?

I have a permutation, say as ${4,1,7,2,3,5,6}$, given by its induced matrix. According to this paper (Proposition 11 on p. 82), To compute its topological entropy, one can compute the ...
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1answer
68 views

Relationship between compression, shannon entropy and kolmogorov complexity

I have read that the Shannon Entropy is used as a bound for the compressibility of a message, for example here 1 it says "In other words, the best possible lossless compression rate is the entropy ...
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1answer
67 views

prove this inequality related to probability and information theory

How do I prove this? I'm thinking I should use Jensen's inequality somehow. $$\sum_K p_k(1-p_k) \le -\sum_K p_k\log p_k$$ The assumption that $\sum_K p_k=1$ holds.
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1answer
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Normalization of data in decision tree

After reading through a few references, I have come to know that for machine learning in general, it is necessary to normalize features so that no features are arbitrarily large ($centering$) and all ...
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1answer
23 views

Tools to compress a finite list as a function

Can someone show me some tool to a lossless compression in an algorithm of a finite list of rational numbers? By example this list A=(0,1,3,2,-1,-2,0), there is a way to construct an algorithm or ...
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1answer
48 views

What is the correct equation for conditional relative entropy and why

I was trying to understand the concept of conditional relative entropy. As in: $$D(P(X\mid Y) ||Q(X\mid Y))= E [\log\frac{P(X\mid Y)}{Q(X\mid Y)}]$$ I would have thought that its equations would ...
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Counting graph isomorphisms and entropy

Question: If all graphs on $n$ vertices are given equal probability, what does the induced probability distribution on the graph isomorphism classes look like? Are there any patterns that emerge as ...
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Entropy derivation from Multiplicity

Multiplicity(W)= N!/(n1!*n2!....ni!) Entropy = 1/N * ln W = 1/N*ln N! - 1/N*sigma_for_all_i(ln ni!) As N->infinity, By Stirlings approximation ...
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1answer
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Meaning of this term:$H(X \oplus\hat{X}|\hat{X} )$

Here, $H$ means the entropy function. I understand that the symbol $\oplus$ means modulo $2$ addition. But I don't understand the significance of the entire expression.
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1answer
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Relation between entropy and compressibility of a file

Suppose I have an ordered list of bytes (the hexdump of some object file), and wish to calculate the information entropy of this file. My understanding is I can calculate this as $$ \sum_{n=0}^{n=255} ...
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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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Measuring the entropy of a graph representing a transition probability matrix of a first order markov chain

There's a research project i'm currently working on which requires me to analyze various aspects of "worlds" represented by transition probability matrices, where the nodes represent objects in the ...
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Finite shannon entropy and mutual information

I was wondering, if it can be shown that the shannon entropy for continuous random variables are finite, such that $H(X)=-\int_{-\infty}^{\infty}f(x)*log(f(x)) dx < \infty$ and the same question ...
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1answer
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Entropy of the product of two random variables

Consider a random matrix $X$ and a random vector $Y$. Let the Shannon entropies $H(X) = H(Y) = n$. Is there a simple upper bound for entropy $H(XY)$? I believe $H(XY) \leq 2n$ as that is a simple ...
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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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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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1answer
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how to find the maximum of the cross-entropy of a discrete random variable?

For a discrete random variable $x$, the cross entropy is $$H(x) = -(p_1\log p_1+\cdots+p_n\log p_n)$$ , so what is the maximum of $H(x)$? Here is what I tried, I compute the gradient as follows ...
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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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1answer
56 views

If a series converges, does it converge with additional log term multiplied?

If $\sum_{n} |a_n| < \infty$, is it true that $\sum_{n} |a_n\log(a_n)| < \infty$ if $0 \leq a_n \leq 1$? I am trying to see if $A$ is trace class operator, then $A \log(A)$ is also trace class ...
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1answer
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InformationGain on Two Continuos classes instead on inary

I've a problem regarding an excersise with information gain. I can't seem to get the right answer, because the excersises differs from what we learned. Usually, a target class is a binary variable ...
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1answer
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Trying to understand the math in a neuroscience article by Karl Friston

I am trying to understand a neuroscience article by Karl Friston. In it he gives three equations that are, as I understand him, equivalent or inter-convertertable and refer to both physical and ...
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1answer
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Finding a specific sequence of digits in pi

Looking at the pifs project on GitHub and this question on SO has made me curious as to how feasible it is to find a specific sequence of digits within Pi. Essentially, on average, how many digits of ...
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1answer
86 views

calculate channel capacity and maximum conditional entropy

i want to know when it is equal channel capacity or $I(X,Y)$ maximum or where $I(X,Y)=H(X)-H(X\mid Y)=H(Y)-H(Y\mid X)$ now if we have two random variable with some specific distribution ...
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1answer
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when it is conditional entropy minimized?

for example let us consider following table know that entropy of variable is maximum when it is equally distributed,all of it's variable has equal probability,but what about joint entropy or ...
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44 views

conditional entropy calculation

let us consider following table i am asked to calculate conditional entropy,from table i have understood everything,for instance how to calculate marginal probabilities,also i know formula for ...
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1answer
35 views

What's the sum of all events? (not the sum of all probabilites of events)

I need to calculate the entropy $h(X|Y)$, where $Y=X^2$. In this case, I suppose $\mathrm p(x|y)=\frac{1}{2}$. For the entropy \begin{align} h(X|Y) &= \int\limits_y \mathrm p(y)\ h(X|Y=y)\ ...
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continuous, non-additive set function

I am looking for a non-additive, continuous set function from a simplex $\Pi_{n}$ of finite dimension $n-1$ into $[0,\infty]$. The motivation is as follows. Shannon defined the entropy ...
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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 ...