0
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0answers
7 views

Are there different types of Shannon's entropy?

The physical entropy (Boltzmann's) is excluded from this question. I ask this question in the purpose of getting more familiar with the concept of informational entropy. To me, it is more of a ...
1
vote
1answer
54 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 $H(S|\hat{S}) \leq K \cdot h(P_e)$ ...
4
votes
0answers
27 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 ...
1
vote
1answer
33 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 ...
0
votes
1answer
34 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 ...
1
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3answers
49 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 ...
2
votes
0answers
28 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 ...
2
votes
3answers
125 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 ...
1
vote
0answers
49 views

Continous wavelet transform and shannon Entropy.

Note: I have asked the same question on signal processing forum,but didn't get any answer. so it might be more like a math or physics question. Hope you don't consider it as cross-post. I am trying to ...
2
votes
1answer
84 views

Estimating the entropy

Given a discrete random variable $X$, I would like to estimate the entropy of $Y=f(X)$ by sampling. I can sample uniformly from $X$. The samples are just random vectors of length $n$ where the entries ...
1
vote
1answer
27 views

Proving the monotonicity of a recurrence.

Define the following recurrence for $n = 1, 2, \cdots$ $T(n) = ( 1 - \operatorname{H}(\frac{1 - P^{\frac{1}{n}}}{2}))^n$ where $0 < P < 1$ is a constant, function $\operatorname{H}(\cdot)$ is ...
1
vote
0answers
48 views

what is relative entropy between to random binary string with length of $L_1$ & $L_2$?

I want calculate relative entropy between two strings of binary such as: $L_1:11000100011101001$ $L_2:00101110110111001$ It is primarily when the lengths of two strings is same and in general when ...
0
votes
1answer
25 views

$Y$ is a function of $X$: making an inference based on the markovity of $ X$

In the information theory book by Cover and Thomas it is written: if $X$ is markov and $Y$ is a function of $X$ then: ...
0
votes
2answers
45 views

Can infinite random sequences be asymptotically compressed?

A number $0.5<p<1$ is chosen at random and given to two people A and B whom are allowed to communicate before beeing separated. A is then given a sequence S of N random bits where each bit has ...
2
votes
1answer
194 views

Graph Entropy - What is it?

I am having trouble getting some intuition as to what graph entropy measures. The definition that I have is that given a graph $G$, $H(G) = \min_{X,Y}I(X ;Y)$, where $X$ is a uniformly random vertex ...
1
vote
1answer
76 views

Notion of Relative Entropy

I do not understand the notion of relative entropy. Relative Entropy. $D_{KL}(P||Q) = \sum_{i}^{}P(i)\log \frac{P(i)}{Q(i)}$. I try to get some intuition why it looks the way it looks. I see that it ...
4
votes
0answers
94 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le ...
1
vote
1answer
243 views

Maximum entropy joint distribution from marginals?

How does one find the maximum entropy joint distribution of two random variables X and Y given their marginal probability mass functions? I know: I have the marginals, meaning p(x) and p(y) are ...
0
votes
1answer
72 views

How much information do you get if you draw a red card?

I'm trying to figure out what this question is asking and what it is I'm trying to calculate exactly. I'm told: You have cards 2-5 of each suit, except the 2 and 3 of the red cards. So 12 cards ...
1
vote
1answer
75 views

Is mutual information transitive?

Suppose A, B and C are random variables. Given that the mutual information between A and B is very large and also the mutual information between B and C is very large, could we conclude that the ...
6
votes
3answers
209 views

What is information theoretic entropy and its physical significance?

I have learned entropy in my information theory classes. The definition I got from text books was the average information content in a message sequence etc. But in one of the MIT videos related to ...
1
vote
0answers
36 views

Calculating entropy of Naive Bayes random variables

Suppose a Naive Bayes graphical model with binary random variables is given by $$P(y,x_1,x_2,...,x_n)=P(y)P(x_1|y)...P(x_n|y)$$ Attempting to calculate $I(x_1,...,x_n;y)$ raises the question: how can ...
1
vote
1answer
193 views

Is maximizing the Shannon differential entropy equivalent to minimizing the predictability and/or minimizing the maximum density?

For a real-valued, 1-dimensional, continuous random variable X with density f(x), I am trying to determine if maximizing the Shannon differential entropy of f(x) is mathematically equivalent to ...
0
votes
1answer
139 views

Information theory entropy equation

I'm studying information theory, and working through this document. On page 17, it shows that, with the function that gets the entropy of a probability $I$ and a probability $p$, that $I(p^a) = a * ...
2
votes
0answers
114 views

Non-zero Conditional Differential Entropy between a random variable and a function of it

Let two continuous random variables, where the one is a function of the other: $X\, $ and $\, Y=g\left(X\right)$. Their mutual information is defined as ...
2
votes
1answer
62 views

Entropy and Shearer's Inequality

I have two questions both related to Shearer's Inequality: 1) When is equality attained in Shearer's Inequality? One trivial instance is when the random variables are all independent. Is this the ...
3
votes
0answers
104 views

Reference for a transformation

Has the (Lebesgue-)ergodic transformation $T: \{0,1\}^{\mathbb{N}} \to \{0,1\}^{\mathbb{N}}$ defined by $T(x(0)x(1)x(2)\cdots) = x(1)x(3)x(5)\cdots$ been well-studied? If so, where? Does it have a ...
2
votes
1answer
86 views

Properties of Entropy

When someone writes $H(X_1, X_2, X_3) = H(X_1) + H(X_2\mid X_1) + H(X_3\mid X_2, X_1)$, how should that last term be interpreted/read? As the joint entropy between 2 variables where variable 1 is ...
0
votes
0answers
63 views

Intregral of exponential of Shannon Entropy Function

Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of $F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$ ...
1
vote
1answer
65 views

Entropy vs predictability vs encodability

Imagine there's a guessing game where a series of binary symbols are presented and a human must decide quickly if the symbol is the same as the previous or different. There's a property of the ...
2
votes
1answer
45 views

Shannon inequalities

I have some difficulties in showing the relationship between mutual information $I(X; Y |Z)$ and $I(X; Y)$? What is larger?
2
votes
0answers
52 views

Computing Relative entropy?

I am doing a project for my CS class and I was wondering if the following would work. I have 50 different people who have rated the same 50 books. The rating system is as follows: negative 5 = hate ...
3
votes
0answers
87 views

Which takes more energy: Shuffling a sorted deck or sorting a shuffled one?

You have an array of length $n$ containing $n$ distinct elements. You have access to a comparator on the elements (a black-box function that takes $a$ and $b$ and returns true if $a < b$, false ...
1
vote
0answers
60 views

convexity of the product of two entropy-like functions

Consider the functions $T_p(q)= \sum_i q_i^p$, where p>1 and q is a finite-dimensional vector satisfying $\sum_i q_i = 1, q_i >0$ (ie, a probability mass function). In information-theoretic terms, ...
0
votes
1answer
43 views

i.i.d binary random variable question

Suppose there are i.i.d. binary random variables $X_i \sim X$ with distribution $P(X=1) = 0.75$ and $P(X=0) = 0.25$ i) For $n=5$ and $e=0.1$, which sequences fall in the typical set $A_e^n$? What is ...
0
votes
1answer
141 views

Entropy Problem: mutual information

I have a problem about entropy and mutual information that I have attempted, but would like feedback on. 30% Boas 20% Anaconda 50% Cobra Half of the Cobras were medium sized, and the other half were ...
1
vote
1answer
73 views

About the differential entropies of well-known continuous distributions

Assume that the continuous random variable $X$ has a distribution (in a closed form expression) with differential entropy $h(X)$. Q) Then, is it true for any continuous distribution that the ...
0
votes
1answer
177 views

How to prove the following entropy formula?

Could anyone show me a proof or redirect to a source where the following entropy equation is proved? =) $$H(X,Y|Z)=H(X|Z)+H(Y|X,Z)$$ Thank you!
7
votes
2answers
252 views

Can the entropy of a random variable with countably many outcomes be infinite?

Consider a random variable $X$ taking values over $\mathbb{N}$. Let $\mathbb{P}(X = i) = p_i$ for $i \in \mathbb{N}$. The entropy of $X$ is defined by $$H(X) = \sum_i -p_i \log p_i.$$ Is it possible ...
0
votes
1answer
154 views

Entropy of a Binary Source with a random until first other result is given.

I was studying for an exam and i found an interesting exercise, but very very bad redacted. A coin is thrown until the first face is found. Denote as X the number of throws required. And find: a) ...
1
vote
0answers
177 views

Mutual Information of Correlated Bivariate Uniform Distribution

We have correlated bivariate uniform distribution, where X and Y have a correlation coefficient $\rho$ and they uniformly distributed in the following rectangle. What is the mutual information of $X$ ...
3
votes
1answer
126 views

Intuition about the relation of combinations and entropy

It is not difficult to show that $${n \choose \lambda n} \leq 2^{H(\lambda)n}$$ where $H$ is the binary entropy function: $$H(\alpha) = -\alpha \lg \alpha - (1-\alpha)\lg (1-\alpha)$$ I was ...
5
votes
2answers
569 views

Inverse of binary entropy function for $0 \le x \le \frac{1}{2}$

I'm trying to find the inverse of $H_2(x) = -x \log_2 x - (1-x) \log_2 (1-x)$[1] subject to $0 \le x \le \frac{1}{2}$. This is for a computation, so an approximation is good enough. My approach was ...
2
votes
0answers
146 views

rényi entropy as a derivative

Let $x=(x_i)$ be a probability measure on $\{1,\ldots,n\}$. Suppose $1<p<\infty$. The Rényi entropy of $x$ is $$ H^p(x)=\frac{1}{1-p}\log \sum_{i} x_i^p. $$ Does there exist a formula for ...
1
vote
1answer
232 views

Explicit examples of smooth entropy computation

Smooth classic entropies generalize the standard notions of entropy. This smoothing stands for a minimization/maximization over all events $\Omega$ such that $p(\Omega)\geq 1-\varepsilon$ for a given ...
2
votes
1answer
190 views

Parameter estimation for a distribution by minimizing its conditional entropy

Let $X$ be a discrete random variable with Laplacian distribution with mean $0$ and scale $\lambda$, as $$ p(X) = \frac{1}{2\lambda} \exp\left(-\frac{|x|}{2\lambda}\right), \\ X \in ...
0
votes
1answer
86 views

How to compute Shannon information?

Given a string of random symbols with yet a priori unknown distribution, what are the known algorithms to compute its Shannon entropy? $$H = - \sum_i \; p_i \log p_i$$ Is there an algorithm to ...
5
votes
1answer
170 views

Lemma in Petersen's *Ergodic Theory*

I'm trying to understand the proof of Lemma 6.2.1 (p.260-261) in Petersen's Ergodic Theory. Specifically, I don't understand why $B_{n}^{A} \in \mathscr{B}(T^{-1}\alpha \vee \dots \vee T^{-n}\alpha)$ ...
1
vote
1answer
98 views

Question about Mutual Information

I am learning about mutual information, and am confused about one of the definitions. Mutual information is defined as $ I(X;Y) = H(X) - H(X | Y) $ where, $$ H(X) = \sum_{x} p(x) \log ...
2
votes
1answer
100 views

Definition of entropy of an ergodic measure

I'm reading a paper in which it is stated that The entropy of an ergodic measure is defined as $$\lim_{n \to \infty} -\frac{1}{n} \sum_{|w|=n} \mu[w] \log \mu[w].\tag{1} \label{eq:1}$$ Here ...