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

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Question about the Shannon Entropy formula

I have a basic question about the Shannon Entropy formula. In fact it's so dumb that I didn't dare ask it in the class because I don't understand the text books. Here's the formula: $$H(X) = -\...
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Mathematical Definition of Entropy and a Question about the Nature of Stat. Mechanics Approach

I have been studying for quite some time now about entropic functionals, including Boltzmann-Gibbs, Renyi, Kaniadakis and Tsallis, and I am familiar with the properties that a functional has to ...
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Decomposition of Shannon conditional mutual information as seen on Wikipedia [Solved]

I am looking for a proof for a formula seen on Wikipedia: \begin{align} I(X;Y|Z) & = H(Z|X) + H(X) + H(Z|Y) + H(Y) - H(Z|X,Y) - H(X,Y) - H(Z) \\ {} & = I(X;Y) + H(Z|X) + H(Z|Y) - H(Z|X,Y) - H(...
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Derivation of joint entropy $H(X,Y) = H(X) + H(Y|X)$

Can somebody explain the calculations with arrows below? And I am sorry if I have placed my post in the wrong place.
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Difference of Entropy of two-dimensional Gaussians

I encountered a putative contradiction. Assume we have two 2-dim. Gaussian variables $z_1 = (x_1, y_1)$ and $z_2 = (x_2, y_2)$ with all components being independent, normal distributed variables: $x_1,...
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Information Channel Capacity

1) Suppose a Noiseless Binary Channel, who's input is reproduced exactly at the output. Let X be the transmitter and Y the receiver (i.e (X=0----->Y=0 and X=1-----> Y= 1)) I understand intuitively ...
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Help me understand the proof for Shannon's Theorem 4 (regarding number of sequences of various probabilities) in the original paper

I'm reading Shannon's 1948 paper where I encountered Theorem 4: $$ \lim \limits_{N \to \infty} \frac {\log n(q)} N = H $$ In Appendix 3 after proving Theorem 3, Shannon proves Theorem 4 by saying: ...
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How to re-estimate a theoretical outcome knowing rng is imperfect

I have 3 regular dice with six sides. I am playing a game where I win when I match all three dice to the same side. My expected shannon bits of entropy H(X) evaluates to 7.754887502 bits. However, my ...
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mutual information and combinatorics

\begin{align} &\mathrm{H}\left(\frac{1}{2^{k}}\right) \\[3mm]&\ \!\!\!\!\!\!\!\!\!\! - {1 \over 2^{k}}\left\{% {k \choose 0}\mathrm{H}\left(\left[1 - \epsilon\right]^{\,k}\right) + {k \choose ...
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Is the Entropy a Function or a Functional? [duplicate]

As in the title, I was wondering whether the entropy of a system (it can be any entropy, from Boltzmann to Renyi etc, it is of no importance) is a function or a functional and why? Since it is mostly ...
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Formula for proportion of entropy

Let's say we have a probability distribution having 20 distinct outcomes. Then for that distribution the entropy is calculated is $2.5$ while the maximal possible entropy here is then of course $-\ln(\...
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Entropy, stirling's approximation

Given that the $\Omega$ is the total theoretical information encoded in a string of characrs let's say, we can express it in the bit-manner: $$\Omega=2^G$$ Therefore $G=\log_2 \Omega$ We know that $...
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18 views

Wave Equation: Distribution which maximises Entropy

Given the wave equation: $\left(\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}\right)f(t,x)=0$ and expanding $f(t,x)$ through a Fourier Transform: $f(t,x)=\int d\omega dk F(\omega,k)e^...
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Definition of Entropy for Homogeneous sample.

In the context of decision tree's we learned that Entropy of a Sample is defined as $H(<P_1...P_n>) = \sum_{i=1}^{n}P_i log_2(\frac{1}{P_i})$ Where $P_i$ is the proportion of the Variable i of ...
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why can't sort 12 elements in 29 comparisons

The information theoretic lower bound for sorting 12 elements is using 29 comparisons, but actually we can't sort them in less than 30 comparisons. My problem is that why we can't reach the ...
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Conceptual Question : Relationship between entropy and a technique for source coding

I want to encode the messages to a sequence of 1s and 0s (subsequently called "bits"). This is called "source coding". Shannon's source coding theory states that the entropy of a source that emits a ...
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How is the entropy of the normal distribution derived?

Wikipedia says the entropy of the normal distribution is $\frac{1}2 \ln(2\pi e\sigma^2)$ I could not find any proof for that, though. I found some proofs that show that the maximum entropy resembles ...
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compute the L^2-distance of a given function to the set of Gaussian functions

I am faced with the following question: given a probability density function $f$ over $\mathbb{R}$ with $\int_{\mathbb{R}}f(x)x^2dx=\sigma^2$ given, I am trying to find the "nearest" Gaussian to $\...
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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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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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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 ((f(X,Y),...
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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 $H(A,B)...
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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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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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(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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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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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 H(X|...
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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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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) = -(.5*...
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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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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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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 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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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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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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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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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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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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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 (\frac{...
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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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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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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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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 p(x,...
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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 x-\...
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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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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 ...