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

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I choose $n$ words from $k$ randoms words from a dictionary with $t$ words. How much entropy is this password?

Let's say I have a dictionary of $t$ words. I randomly select a set of $k<t$ words (no duplicates). Next, I deterministically choose $n<k$ words from those $k$ words (say, pick the first $n$ ...
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23 views

Correlation and entropy between stocks of the same index

My portofolio contains the stocks belonging to Nasdaq100 index. Initially, i found the entropy of closing prices between Friday's value and Monday's value,for all companies of the index, in order to ...
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Asymptotic binomial ratios

I am in need of asymptotic version of $$\frac{ \displaystyle \binom{n^{1-s}}{n^s}}{\displaystyle \binom{n}{n^{s}}}$$ where $n\in\Bbb N$ and $s\in\big(0,\frac12\big)$ and $$\displaystyle \frac{ ...
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compute H(X|Y) ( conditional probablity)

Can someone help me on this? X = {$X_1,X_2,X_3,X_4$} Y = {$Y_1,Y_2,Y_3,Y_4$} Suppose p($X_i$) = p($Y_j$) = 1/4 (each X and each Y equally likely) $1 \leq i, 4 \ge j$ and now suppose $Y_1 : X_1 ...
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definition of conditional probability $(p_i|\pi_k)$ and Tsallis entropy

Let $\Omega$ be a set of $W$ possible outcomes of an experiment with probability assignments $p_i$ and thus $\sum_{i=1}^{W}p_i=1$. Now, let's divide $\Omega$ into $K$ non-intersecting subsets each ...
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39 views

Proof of chain rule for entropy of random variables

I have the following proof for the chain rule for entropy of random variables: We write: \begin{eqnarray*} ...
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32 views

Mutual information between 2 sequences of random variables?

How would I go about expanding $I(X_1,...,X_n;Y_1,...,Y_n)$? The chain rule exists for a single case, i.e.: $I(X_1,...,X_n;Y)=\sum^n_{i=1} I(X_i;Y|X_{i-1},...,X_1)$, but I'm having doubts if this can ...
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How to evaluate the quality of the probability distribution output of a classifier?

In a classification problem, I have trained a neural network which outputs class probabilities for a given input. For a new input, I now want to evaluate the "quality" of the neural network's ...
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51 views

Proof of Central Limit Theorem via MaxEnt principle

Let $X_i$'s be i.i.d. with mean $0$ and variance $\sigma^2$. After reading Jaynes' book: Probability the Logic of Science, I decided to try out and actually prove CLT via the following steps: a) ...
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Is it possible to code with less bits than calculated by Shannon's source coding theorem?

In information theory, Shannon's source coding theorem establishes the limits to possible data compression, and the operational meaning of the Shannon entropy. Consider that we have data generated by ...
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45 views

Conditional joint information of two random variables $X,Y$ given $Z$

For 3 random variables I am trying to prove the following: \begin{eqnarray*} I(X;Y|Z)&\triangleq& H(X|Z)-H(X|Y,Z)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) ...
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Binary Symmetric Channel with Feedback

Suppose that feedback is used on a binary symmetric channel with parameter p. Each time a Y is received, it becomes the next transmission. Thus $X_1$ is Bern(1/2), $X_2$ = $Y_1$; $X_3$ = $Y_2$,..., ...
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36 views

mutual information entropy problem

In mutual information we have: if $x$ and $y$ are independent then $p(x,y)=p(x)p(y)$ and then $I(X;Y)=0$. Do If $I (X;Y) = 0$ when $x$ and $y$ are not necessarily independent?
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Estimate password entropy by “trying out” passwords

The entropy of a password of a fixed length $n$ and $c$ possible characters is calculated by $n*log_2(c)= log_2(c^n)$, see also here: ...
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148 views

Is entropy of prime numbers smaller?

Seems that entropy (in information theory) can be expressed as a measure of how unpredictable is each bit of information. I have done a little experiment: I've measured entropy of the binary ...
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49 views

Conditional joint entropy of two random variables

I am trying to prove the formula that gives the joint entropy of the random variables $X$ and $Y$ given $Z$ which is: $$H(X,Y|Z) = H(X|Z) + H(Y|X,Z)$$ based on the definition of conditional entropy ...
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71 views

Entropy of union of multisets

Assigning a random variable to some multiset: Assume that $S$ is a multiset. We can think of $S$ as independent sampling from some random variable. For instance, $S = \{H, H, T, T, T\}$ can be thought ...
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How can I compare the entropy of two vectors with different sum?

I wanna compare the entropy of two vectors, e.g. [1,2,3] vs [0.2, 0.1, 0.3]. Note that the two vectors have different sum, i.e. 6 and 0.6. Basically I wanna know which vector is more unbalanced. ...
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172 views

Jensen's inequality proof

The standard proof for Jensen's inequality using taylor expansion around a point $x_0$ involves using only first 3 terms of the Taylor series till $f^{\prime \prime}(x)$. Why are we able to ignore the ...
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261 views

An upper bound of binary entropy

Binary entropy is given by $$H_{\mathrm b}(p) = -p \log_2 p - (1 - p) \log_2 (1 - p), \hspace{6 mm} p \le \frac{1}{2}$$ How can I prove that $$H_{\mathrm b}(p) \le 2 \sqrt{p(1-p)}$$
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28 views

Prove entropy theorem

Doing a course of cryptography I have been asked to prove the following: $H(X,Y) = H(Y) +H(X|Y)$. But I simply do not know where to start, so a hint in the right direction would be very much ...
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31 views

Open covers and $(n,\varepsilon)$-separating/ spanning sets: proving three inequalities

In Peter Walters' book An Introduction to Ergodic Theory, one can find the following corollary (p. 174 in my edition). At the end of this thread, I tried to prove it. It would be great if you could ...
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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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Is the topological entropy of a continuous map $T\colon X\to X$ zero if $X$ is a finite topological space?

Let $X$ be a finite topological space and $T\colon X\to X$ continuous. As the title already suggests, I am wondering if the topological entropy of $T$, denoted by $h(X,T)$, then is $0$. As far as I ...
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Source coding with 2 distinct distributions and entropies

I'm learning about source coding, and many of the books/resources I've read give examples of the source $X^n$ being defined as a sequence of iid random variables. How about when the sequence is ...
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139 views

Very special geometric shape (No name yet?)

I suppose this geometric shape is something very 'special'. I cannot clarify in short about being 'special', but I think this shape stands together with such special shapes like the square and the ...
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38 views

a problem involving binary entropy function

let $\alpha<1/2$ such that $2^{H(\alpha)}\le 2^{1-\epsilon}$,when $H$ is binary entropy function. how can i prove that then we have: $2^{n(1-\epsilon)}\ge \sum\limits_{i\le \alpha n } {n \choose ...
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92 views

Definition of topological entropy

What the meaning of the limit that appears in the definition of topological entropy? Let $X$ a compact metric space and $f\colon X\to X$ a continuous function and a subset $K \subset X$. The ...
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267 views

Calculating Shannon Entropy for DNA sequence?

I'm following the formula on http://www.shannonentropy.netmark.pl/calculate to calculate the Shannon Entropy of a string of nucleotides [nt]. Since their are 4 nt, ...
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173 views

Zero conditional entropy

This question is related to this math.se question but I need a bit more guidance. For two discrete random variables $X,Y$ we define their conditional entropy to be $$H(X|Y) = -\sum_{y \in Y} Pr[Y = ...
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46 views

Is mutual information convex in the joint distribution?

Assume some joint distribution $P(X,Y) = P(Y|X)P(X)$. It is well know that, for fixed $P(Y|X)$, mutual information is a concave function of $P(X)$ and, for fixed $P(X)$, a convex function of $P(Y|X)$ ...
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Concentration property of entropy

Let $X$ be a random variable taking its values in $A = \{a_1,\ldots,a_n\}$ such that $Pr[X = a_i] = p_i$ for all $1 \leq i \leq n.$ The entropy of $X$ is defined as $$H(X) = -\sum_{i=1}^n p_i ...
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Differential Entropy

I'm a little temporarily confused about the concept of differential entropy. It says on wikipedia that the differential entropy of a Gaussian is $\log(\sigma\sqrt{2\pi e})$. However I was thinking as ...
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159 views

Is the Library of Babel random? Does it contain information?

The Library of Babel is defined as a universe in the form of a vast library containing all possible 410-page books of a certain format and character set. However, applying two means of ...
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Why $H_{V^* \cup W^*} > H_{V \cup W}$ if $H_V$ denotes entropy of language

Let $W \subseteq X^*$ be an infinite language over a finite alphabet $X$, and define ($|w|$ denotes the length of $w \in W$) $$ H_W := \limsup_{n\to \infty} \frac{\log_{|X|} | \{ w \in w \in W, |w| = ...
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Shannon information for a set of 3 equally probable elements?

How to calculate entropy as a number of binary choices for a set of three equally probable elements? The Shannon's formula gives $\log_2(3)=1.585$. But any interpretation of binary choices gives me ...
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68 views

Entropy of $f(x)=1$

Let $f(x)$ be a probability density function $f(x) = 1$ on $x = [0,1]$, and entropy defined as $$H(p(x)) = -\int p(x) \log_2(p(x)) \, dx$$ where $p(x)$ is a pdf. Unless I've made an arithmetic error, ...
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54 views

Approximation for a binomial coefficient sequence summation

What is a good approximation to $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{\binom{k(k-1)/2}{i}}$$ $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{(\log ...
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Mutual Information: Are these two equations equal?

I'm working with Multivariate Mutual Information (MMI), specifically with three variables $(X,Y,Z)$, applied to RNA sequences. The MMI equation that I use for three variables is based on entropy ...
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88 views

Partition-based entropy of a sequence

The entropy $H$ of a discrete random variable $X$ is defined by $$H(X)=E[I(X)]=\sum_xP(x)I(x)=\sum_xP(x)\log P(x)^{-1}$$ where $x$ are the possible values of $X$, $P(x)$ is the probability of $x$, ...
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How to efficiently select a subset of elements that maximizes a certain property? (entropy)

I need to select $k$ elements from a pool containing a much larger number $N$ of elements. The selection must be done in a way that a function $h(\{z_{i_1},\ldots,z_{i_k}\})$ is maximized or ...
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How to calculate entropy from a set of samples?

entropy (information content) is defined as: $$ H(X) = \sum_{i} {\mathrm{P}(x_i)\,\mathrm{I}(x_i)} = -\sum_{i} {\mathrm{P}(x_i) \log_b \mathrm{P}(x_i)} $$ This allows to calculate the entropy of a ...
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“Empirical” entropy.

Information entropy is usually defined as $$\text{I}_b({\bf p}) = -\sum_{\forall i}p_i\log_b(p_i)$$ i.e. the expected value of the negative logarithm of the probabilities. This is all good when we ...
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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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How does a wavelet help in compressing data

I have an understanding of how we carry out image compression by using DCT along with Huffman encoding. The next subject is wavelets. I understand that wavelets are small waves and there are different ...
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168 views

Shannon Entropy Maximization with Constraints

I have got a cumulative distribution function $F_X(x)=Pr(X<=x)$. This distribution is described by 2 parameters $\alpha, \beta$. We define $F_k$ as follows: $\forall k<=n_k, ...
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185 views

Fano's Inequality Proof

For an information theory class, I am studying the proof for Fano's inequality, i.e.: $H(P_e) + P_elog(|X|) \geq H(X|\hat{X}) \geq H(X|Y)$ Where $H(X)$ is the entropy of the random variable X ...
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Mutual Information: How these two equations are equal?

I'm a biologist trying to apply the Mutual Information (MI) to some RNA secondary structure. I know that there exists two MI equation that, mathematically, are equal: $I(X,Y) = \sum_{x,y} p(x,y) ...
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Infinite closed shift-invariant set of binary sequences with zero entropy?

Does there exist an infinite closed shift-invariant $X \subset \{0,1\}^\Bbb Z$ with zero topological entropy? How to think of an example? Will periodic points of shift$|_X$ have zero entropy?
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The difference in entropy rates between a hidden process and its observation

Let $S$ be a finite state space and $o:S\to S$ an observation function. Given a distribution $p$ on $S\times S$, consider the following optimization problem: $$\max \left[ EntropyRate(\{x_t\}) - ...