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

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Nearly a factor map (only shifted)

For topological entropy, we have that $h(T)\geq h(S)$ if $(Y,S)$ is a topological factor of $(X,T)$, i.e. $T=\phi S\phi^{-1}$ where $\phi\colon X\to Y$ is a continuous surjection. That is we have ...
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Rate distortion function with infinite distortion

I am working through the problems in Elements of Information Theory by Cover and Thomas and have come across the following problem I couldn't answer. The problem is to find the rate distortion ...
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1answer
21 views

Differential entropy of $\Gamma$

Let $X \sim Gamma(\alpha,\beta)$ be gamma distributed random variable with probability distribution function $$ f_{X}(x)=\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)},\;x>0 $$ ...
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Looking for a a measure-theoretic treatment of “differential entropy”

If $X$ is a discrete random variable, its entropy $H(X)$ is usually defined as something along the lines of $-\sum \def\P{\mathbb{P}}\P(x) \log_2( \P(x))$, where the sum ranges over all the possible ...
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(Elegant) proof of an inequality: $h(x) \geq 1- (1-\frac{x}{1-x})^2$, where $h$ is the binary entropy function

I am looking for the most concise and elegant proof of the following inequality: $$ h(x) \geq 1- \left(1-\frac{x}{1-x}\right)^2, \qquad \forall x\in(0,1) $$ where $h(x) = x \log_2\frac{1}{x}+(1-x) ...
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1answer
30 views

Comparing entropies $H((f(X,Y), g(X,Y)))$ and $H ((f(X,Y),g(X,Z)))$

Let X,Y,Z be three independent uniform distributions on $\{0,1\}^n$; $f, g:\{0,1\}^n\times\{0,1\}^n\rightarrow\{0,1\}$ be two boolean functions. Is it true that $$H((f(X,Y), g(X,Y)))\leq H ...
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Entropy of a binary string obtained from dynamical system and length of the source code

H(S), the entropy of a source, gives you the average codeword length to encode a given source alphabet. i.e. it is the average number of bits per symbol required to encode the information in the ...
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Conditional entropy $H(A,B|C)$ independent of whether $p(A,B)$ factorises or not

I arrived at a (slightly different but hopefully more precise) reformulation of my original question shown below: Is $H(A,B)-H(A,B|C)$ maximal for independent $A$ and $B$ in a similar sense as ...
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Does, S = k ln W == W = e^s/k? [closed]

"Boltzmann's equation relates the entropy S of an ideal gas to the number W of microstates corresponding to a given macrostate, via the equation S = k ln W where k is the so-called Boltzmann ...
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Anti-intuition when finding statistical model of a random variable using Maximum Entropy Principal

I was trying to understand the Maximum Entropy Principal, and was calculating a very simple example, but ran into some confusion. Consider a random variable $X$, which can only take values $1,2$ and ...
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Calculating change in entropy, Integration help?

So I need to calculate the change in entropy for a non-ideal gas over two states. The equation im looking at using is $$T\,ds = dh - v\,dP \text{ or } T\,ds = du + P\,dv$$ where $h=u+Pv$ , $s = ...
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3answers
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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1answer
48 views

Is there a statistical measure of bitwise entropy?

(Somewhat inspired by this website, particularly Section III. Also, I might be using a different definition of entropy than usual; what I am using is closest to the physics definition (the one I ...
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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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489 views

Notation of cross entropy

I have a question regarding a notation that seems to be very usual. For starters, cross entropy is defined by: \begin{align}H(X, q) &= H(X) + D(p||q) \\ & =-\sum_x p(x)\log_2 q(x)\end{align} ...
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Change of bases for entropy

From Cover and Thomas, Elements of Information Theory: Why isn't it: $ \log_b(p) = \frac{\log_a(p)}{\log_a(b)} $, so that $ H_a(X) $ is multiplied with $ \frac{1}{\log_b a} $?
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Entropy of a 2-dimensional function versus 1-dimensionl function.

I am a novice in information theory so this is more of a question seeking pointers to ideas/references to think further on the thought. I want to make concrete the idea that a function of two ...
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1answer
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Does the Information Gain algorithm favor a high-entropy attribute or a low-entropy one?

This might not be mutual to mathematics but it does relate to Information-Theory. My question is: Does the InformationGain algorithm, in Decision-Tree machine-learning, favor a high-entropy ...
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1answer
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Simplyfying Shannon Entropy formula

I need to calculate the following which is Shannon Entropy formula only using simple functions like log or squareroot or I don't know, just make it simple enough for a guy that does understand only ...
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1answer
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Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
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Derivation of the maximum entropy distribution

I am reading a book and having trouble following something. The problem is to try to maximize the differential entropy $-\int_{0}^{\infty}p(r)\log p(r)$ under the constraints that ...
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1answer
57 views

Conditional entropy under quantization

Let $X$ be a continuous random variable and $X^n$ its quantization that becomes finer with larger $n$. Let $Y$ be a deterministic function of $X$. Then we have that the conditional entropy $$H(Y|X) = ...
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1answer
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proof of upper bound on differential entropy of f(X)

I asked a similar question yesterday, but I organized my question here a little and further asked my second question. Suppose $X$ is a continuous random variable with the pdf $f_x$, and $Y=g(X)$. If ...
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How to test quality of probability estimates?

I have a Markov chain model which produces a probability distribution for absorption in 4 possible absorbing states. I.e. the model estimates the probability distribution for a discrete random ...
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Channel capacity of sum of symmetric channels

I've got a channel matrix $P$ of the form $\begin{bmatrix} Q \\ R \end{bmatrix}$ where $Q,R$ are channel matrices of symmetric channels, so they now have different input alphabets but the ...
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1answer
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Basic Entropy Inequality and Identity question

This is a solution to a problem I am working on: \begin{equation} \begin{aligned} H(X|Y) + H(Y|Z) &\ge^? H(X|Y, Z) + H(Y|Z) \\ &=^\text{?}H(X,Y |Z) \\ &= H(X|Z) + H(Y|X, Z)\\ &\ge ...
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Entropy of degenerate multivariate normal

After having read the Wikipedia entry on the degenerate case of the multivariate normal distribution: https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case my question is: ...
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How to calculate the Shannon Entropy for a block length of a word

I have a binary sequence of length N as $10110110111...$ I want to segment the above series into equal blocks of a window of length $L$. One way of determining the block length is using the ...
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1answer
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Calculating Entropy and Information Gain of a Variable

I have the following values for two random variables. I need to compute the following values: a. H(Y) b. H(Y|X) c. and finally IG(Y|X) I will show what I have calculated so far. a. H(Y) = ...
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Estimation for entropy

Let $T\colon X\to X$ be continuous and $X$ compact and $K\subset X$ compact. By $s_n(2^{-k},K,T)$ denote the maximal cardinality of any $(n,2^{-k})$ separated subset of $K$. Suppose, we know for ...
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Calculating mutual information for a dataset

I have a dataset of individual text documents $D = {d_0, d_1, ..., d_n}$ and a corpus of keywords $K = {k_0, k_1, ..., k_m}$ in the documents. There are zero or more keywords in each text document. I ...
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von Neumann entropy and change of basis

The von Neumann entropy is defined as $S(\rho)=-Tr({\rho \ln \rho})$, where $\rho$ is density matrix. http://en.wikipedia.org/wiki/Von_Neumann_entropy In the above article it says: S(ρ) is ...
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pattern-sensible entropy measure

I have some binary images (meaning each pixel can be 0 or 1), I want to find a pattern-sensible entropy measure, which means for example that a chessboard should have a very low entropy value (almost ...
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Can the entropy of a system of multiple stochastic processes be defined?

I'm trying to measure the entropy of BINGO cards. (waiting for laughter to die down.) Normally one would think of this as 75 choose 24, but in reality it is 15 choose 5, four times, and 15 choose 4, ...
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183 views

Is there a possibility to determine/ estimate the topological entropy?

By $E$, denote the set of excited states $E=\left\{1,2,\ldots,e\right\}$ and by $R$ the set of refractory states $R=\left\{e+1,e+2,\ldots,e+r\right\}$. By $0$, denote the equilibrium state. The ...
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Calculating the entropy of original set E(S)

I am currently revising for an exam and I am struggling with this question Using a decision tree algorithm with information gain splitting, which of the two attributes diarrhea or fever is a ...
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Entropy of factor map

Let $A$ and $B$ be two compact metric spaces with $B\subset A$- Moreover, let $T\colon A\to A$ continuous and let $S\colon A\to B$ a continuous surjection with $S\circ T=T\circ S$. Moreover assume ...
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Compare two distributions with varying focus on different regions

I have been trying to find if my problem matches has been discussed in prior research and if any technique exists to solve it. Here's the problem: Given two distributions (pdf) D1 and D2 over a ...
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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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Calculating Split Info with given equation not matching solution

I'm given the formula to calculate the Split info but cannot seem to calculate the correct answer of 0.926 that the example shows. Split Info $= -\Sigma \frac{\mid D_j\mid}{\mid D\mid} * log_2 ...
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Deriving the q-Gaussian PDF

Ok it may sound a bit too simple but I am quite confused here. While studying generalized entropic forms, in my case that of $S_q$ or in another words the Tsallis Entropy, I reach a point where I have ...
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1answer
21 views

Decomposition of the entropy

So, I'm reading about this property in the MacKay book. But I don't fully get it. Can someone explain it to me? There's this example: A source produces a character $x$ from the alphabet $A = \{0, ...
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Conditional Entropy and Gibbs Inequality

We know $$H(X | Y) + H(Y) = H(X, Y)$$ Therefore, $$H(X | Y) \leq H(X, Y) $$ since $$ H(Y) \geq 0$$ If we expand this out, we get $$-\sum_{x,y} {p(x,y) \log p(x | y)} \leq - \sum_{x,y} {p(x,y) \log ...
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Entropy/Variance inequality

The following inequality is sometimes used as a building block to prove log Sobolev inequalities. Does anyone have a simple proof of it? $$ x\log x + y\log y - (x+y)\log \frac{x+y}{2}\leq (\sqrt ...
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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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Convert a joint entropy matrix to a contidional entropy matrix.

I've only barely started to learn about Entropy and Information Theory as a part of a course I'm taking in Systems Theory / Cybernetics. The thing is, I'm terrible at math! Say I have a joint ...
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Proving that the entropy is zero given conditional entropies

Let's suppose we have 4 random variables $X,Y,Z$ and $T$ and that the following equations hold about the entropy: $$H(T|X)=H(T)$$ $$H(T|X,Y)=0$$ $$H(T|Y)=H(T)$$ $$H(Y|Z)=0$$ $$H(T|Z)=0$$ Also, the ...
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Does entropy inequality hold for convex combination

I have two pairs of Random Variables, $(\mathbb{X},\mathbb{Y})$ and $(\mathbb{M},\mathbb{N})$ which satisfies, $H(\mathbb{X})>H(\mathbb{Y})$ and $H(\mathbb{M})>H(\mathbb{N})$. For some convex ...
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Password entropy of famous xkcd comic

The famous xkcd comic about password strength calculates the entropy of the 11-character password "Tr0ub4dor&3" with 28 bits of entropy. When following the ASCII-95-chart, we have 95 possible ...
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Existence of measure(s) of maximal entropy, given a finite-to-one chaotic global attractor $A$ which is, moreover, the non-wandering set

Let $X$ be a compact metric space and let $f\colon X\to X$ define some dynamics, $f$ being continuous and finite-to-one on a global attractor $A\subset X$. Moreover, $A$ is the non-wandering set ...