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

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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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250 views

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 ...
2
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1answer
62 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
15 views

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
44 views

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 ...
6
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1answer
250 views

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
265 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
52 views

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 ...
0
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1answer
37 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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0answers
41 views

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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2answers
96 views

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 ...
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1answer
311 views

How is the formula of Shannon Entropy derived?

From this slide, it's said that the smallest possible number of bits per symbol is as the Shannon Entropy formula defined: I've read this post, and still not quite understand how is this formula ...
0
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1answer
35 views

Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$. Is it true that: If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...
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1answer
32 views

Gaussian distribution variance estimation

It's well known if I have a process generating normally distribuited data, I can estimate the parameters of the gaussian function: ...
5
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1answer
64 views

Prove that bitstrings with 1/0-ratio different from 50/50 are compressable

I'm looking for a proof, that $$ \sum_{i=0}^{\lambda n} \binom{n}{i} \le 2^{nH(\lambda)} $$ with $n>0$, $0 \le \lambda \le 1/2$ and $ H(\lambda)=-[\lambda log \lambda + (1-\lambda) log (1-\lambda)] ...
5
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1answer
176 views

Entropy of the induced transformation

I need help with this problem: Let $(X,\mathcal{B},\mu,T)$ be a ergodic dynamical system in the probability space $(X,\mathcal{B},\mu)$. Let $A \in \mathcal{B}$ with $\mu(A)>0$. We define the ...
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1answer
81 views

Variational Methods, why KL divergence is the difference between true distribution and approximating distribution.

Likelihood = $L(\textbf{w}) = P(V\mid \textbf{w})$. $$\ln P(V\mid \textbf{w}) = \ln \sum_H P(H,V\mid \textbf{w})$$ $$= \ln \sum_H Q(H\mid V)\frac{P(H,V\mid \textbf{w})}{Q(H\mid V)}$$ $$\geq ...
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0answers
71 views

What connections between computer science and ergodic theory?

I have a background in ergodic theory, but I am switching to computer science/programming. I would like to know if tools from ergodic theory could be useful, especially something around coding of ...
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1answer
83 views

Entropy of sum is sum of entropies

Having $X$ and $Y$ discrete random variables above finite set. Z is defined as $Z=X+Y$ when does the following happen: $$H(Z)=H(X)+H(Y)$$
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28 views

intuition for entropy solutions

For a hyperbolic PDE of the form $$u_t + f(u)_x = 0$$ it turns out that the right notion of solution is entropy solution. Now, the notion of classical solutions are obviously very natural, and also ...
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2answers
123 views

Entropy of sum of two Uniform random variables

say $X$ and $Y$ are two identical, independent and discrete Uniform random variables and $Z=X+Y$. I do not know more about the random variables. Assuming $H(\cdot)$ to be the entropy of a random ...
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1answer
49 views

Prove identity involving the Tsallis q-logarithm

The natural logarithm and the exponential can both be generalized to a called q-logarithms and q-exponentials.those functions are defined as follows: \begin{eqnarray} \log_q(x) &:=& ...
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0answers
51 views

What is the “true” entropy of a binary string?

Consider an infinite binary string $\sigma$ and define its entropy $$H_1 = -(p_0 \log_2 p_0 + p_1 \log_2 p_1)$$ with $p_i = \lim_{N\rightarrow \infty} N(i)/N$, $N(i)$ the number of $i$'s among the ...
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53 views

Problem with calculating probability of symbols

I've a $100 \times 100$ binary matrix it`s constructed with this probability table : i want to apply extended Huffman on this matrix my idea is to compress each column individually . - so starting ...
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25 views

Can we calculate entropy for nonstationary random variables?

Let's assume that $X$ is a discrete random variable, which can take any value from the set $\{x_0,\dots,x_n\}$ with the probability mass function $P(X)$. We can calculate the entropy of $X$ as ...
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1answer
45 views

Generlized Entropy compared to Generalized Dimension

I am currently reading the following paper by F.Takens: Multifractal analysis of dimensions and entropies. This paper discusses two different measures. One is generalized entropies and the other is ...
3
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1answer
63 views

necessity of uniform continuity for topological entropy

I am referring to Walters' book "Introduction to Ergodic Theory." When he defines the concept of topological entropy he always assumes that the transformation $T: X \rightarrow X$ is uniformly ...
3
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0answers
87 views

Huffman codes: does less entropy imply less weighted average codeword length?

Let $\Sigma$ be a source alphabet with a probability distribution over its symbols $P$. Then, the Shannon entropy of $\Sigma$ is $$-\sum p_j \times -\mbox{log}_2(p_j)$$ where $p_j$ is the probability ...
0
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1answer
57 views

Subadditivity of Entropy

We define $H(X) = -\sum_{x}p_{x}\log p_{x}$ and relative entropy as $H(p(x)||q(x)) = \sum_{x}p(x)\log \frac{p(x)}{q(x)} = -H(X)-\sum_{x}p(x)\log q(x).$ Now we have to prove that $H(X,Y,X) + H(Y) \leq ...
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2answers
100 views

Understanding conditional entropy intuitively $H[Y|X=x]$ vs $H[Y|X]$

I was trying to understand conditional entropy better. The part that was confusing me exactly was the difference between $H[Y|X=x]$ vs $H[Y|X]$. $E[Y|X=x]$ makes some sense to me intuitively because ...
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1answer
32 views

Are measures of information model specific?

Does an information measure for a signal do a better job if it assumes some things about the signal? For example: I have a digital stream of data, 0s and 1s coming at a clock rate $r$. What is the ...
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0answers
43 views

Solomonoff induction , Shannon Entropy, Kolmogorov Complexity.

If Expected Kolmogorov Complexity equals Shannon Entropy why can't Shannon Entropy be used as an approximation of Kolmogorov Complexity in Solomonoff Induction? Regarding Kolmogorov Complexity and ...
3
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1answer
86 views

toplogical entropy of general tent map

Measure theoretic entropy of General Tent maps The linked question made me wonder how to calculate the topological entropy of a general tent map. Let $I=[0,1]$ and $\alpha \in ( 0,1)$. Define $T: I ...
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2answers
143 views

Fano's inequality and error rate

The Wire-tap channel II (http://link.springer.com/chapter/10.1007%2F3-540-39757-4_5) article in proof of Theorem 1 uses Fano's inequality to estimate the entropy $H(S|\hat{S}) \leq K \cdot h(P_e)$ ...
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77 views

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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2answers
80 views

Shannon Entropy Minimization

The Shannon Entropy for an observation is given by $ -x \log_2(x)$. Why is the maximum entropy achieved at $x = \frac{1}{e}$, and not at $x = 0$? Could someone provide a logical explanation that ...
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3answers
266 views

probabilistically what can we say about the next throw of a coin after n throws

this may sound easy or hard or whatever but i cant seem to find anything after searching around for a similar question/answer The question is this: What can we say (probabilistically) about the next ...
0
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1answer
46 views

For P0 close to P1 the relative entropy can be approximated by its series expansion,Why?

I am reading a article (An overview of distinguishing attacks on stream ciphers, Martin Hell · Thomas Johansson · Lennart Brynielsson) about Distinguishe Attacks. There is a approximate equation ...
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59 views

Maximize the expected Brier Score of P relative to P.

Fix a finite set $\mathcal S$ and let $\mathcal P$ be the collection of probability functions over the Boolean closure of $\mathcal S$. Let $\beta : \mathcal P \times \mathcal S \to \mathbb R$, with ...
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73 views

Decrease of entropy when iterating a random discrete function

Let $m$ be a positive integer. Let $S$ be the set of non-negative integers $x$ less than $m$, with $|S|=m$. Let $X_0$ be the discrete uniform distribution over $S$, with $P(x)=\begin{cases} 1/m & ...
2
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1answer
2k views

Correct algorithm for Shannon entropy with R

Shannon entropy is defined by: $H(X) = -\sum_{i} {P(x_i) \log_b P(x_i)}$, where b could be $e$, 2 or 10 (bit, nat, dit, respectively). My interpretation of the formula is: $H(X)$ is equal to the ...
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3answers
66 views

Does entropy $H(y)$ decrease as $H(x,y)$ decreases when $H(x)$ is fixed?

Can't find any proof in Shannon's 1948 paper. Can you provide one or disproof? Thank you. P.S. $H(x)$(or $H(y)$) is the entropy of messages produced by the discrete source $x$(or $y$). $H(x,y)$ is ...
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2answers
104 views

Wrong result from LLR using Dunning Entropy method

I'm trying to use Dunning's method of calculating LLR to compare word instances between two fulltext indexes. His method uses entropy as part of the calculation. Dunning's blog post: ...
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1answer
80 views

Topological Entropy of $(x,y)\mapsto(x+y,x+a)$

Let $a\in \mathbb T^1$, how can I calculate the topological entropies of the maps $T_1:(x,y)\mapsto(x+y,x+a)$ and $T_2 : (x,y) \mapsto (x+y,y+a)$ defined on $\mathbb T^2$. Here $\mathbb T^n$ is the ...
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2answers
41 views

Interpreting Password Entropy Calculation – Property of Character Entropy

I was reading this explanation on how to calculate the entropy of a password. The article is great and it explains it very succinctly that even I understood it. According to the site, if you have a ...
3
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0answers
89 views

Topological applications of topological entropy

I just learned topological entropy during a lecture about dynamical systems, and I wonder whether there exist purely topological applications of it.
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49 views

Joint entropy maximization with a constraint

So I have 2 random variables X and Y, where X can take on values {0,1,2,3} and Y can take on values {0,1,2,3,4}. I need to maximize H(X,Y) subject to the constraint that P(X≠Y)=0.5. This also gives ...
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3answers
862 views

Is Standard Deviation the same as Entropy?

We know that standard deviation (SD) represents the level of dispersion of a distribution. Thus a distribution with only one value (e.g., 1,1,1,1) has SD equals to zero. Similarly, such a distribution ...
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1answer
735 views

How is the entropy of the multivariate normal distribution with mean 0 calculated?

Here is what I have so far: $$\begin{align} h(x) &= - \int \frac{1}{(2\pi)^{\frac{D}{2}}\det\Sigma^{\frac{1}{2}}} \exp(-\frac{1}{2} x^T\Sigma^{-1}x) \ln ...
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3answers
90 views

Roll a dice, ignore result if result is maximum value

Let's say I have a six-faced dice, but I only want results between $1$ and $5$. One way to do that would be to roll a ten-faced dice and divide the result by two ($1-2$ becomes $1$, $3-4$ becomes ...