10
votes
2answers
156 views

Intrinsic significance of differential entropy

Many commentators (e.g. Jaynes, Rota) argue that the notion of "differential entropy" is problematic (as commonly defined by $ h(X) = \int ( \log\frac{1}{p(x)} ) p(x) \, dx $, where $X$ is a random ...
3
votes
1answer
123 views

General additive function of probability

Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that: $H$ is continuous, $H$ is symmetric w.r.t. the order of its arguments, ...
5
votes
0answers
170 views

Maximizing Renyi entropy for a certain channel

The channel under consideration is $T = A + B$, where $A$ and $B$ take on values in $\{0, 1\}$ according to a probability mass function. Let (joint) random vector $(A_1, A_2,\ldots, A_n)$ be denoted ...
3
votes
1answer
365 views

Simple reason that a mathematician cannot do better than random when guessing contents of a box?

I have a question about the finite analog of the puzzle proposed here involving mathematicians guessing the contents of boxes. Specifically, suppose there are $k$ unopened boxes each containing a ...
15
votes
1answer
664 views

Gini Coefficient and Renyi Entropy

Gini coefficient (aka Gini Index) is a quantity used in economics to describe income inequality. It is 0 for uniformly distributed income, and approaches 1 when all income is in hands of one ...
2
votes
0answers
84 views

order of convergence of the conditional entropy (3)

I'm sorry for having open two questions which have been solved by elementary counter-examples provided by @AnthonyQuas. Actually I'm not an expert in information theory and I expected that a positive ...
1
vote
1answer
93 views

order of convergence of the conditional entropy (2)

Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g_n(X_n)$, then by Fano's inequality ...
16
votes
1answer
838 views

An Entropy Inequality (generalized)

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability ...
29
votes
1answer
2k views

An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
4
votes
3answers
435 views

Incremental entropy computation

After a quick internet search I found no method for incremental entropy computation. Question 1 Let $\{x_i\}_{i=1}^n$ and $\{x_i\}_{i=1+n}^{n+m}$ be two samples and let $S_i^j:=\sum_{k=i}^j x_k$. ...
9
votes
1answer
542 views

Entropy of edit distance

The edit or Levenshtein distance between two strings is the minimum number of single character insertions, deletions and substitutions to transform one string into another. If we take random binary ...
1
vote
1answer
252 views

Calculate channel capacity of general channel under constraint

Hi! Given a conditional distribution $P_{Y|X}$ I'd like to find the prior distribution $P_X$ that maximizes the mutual information $I(X;Y)$ with $P_Y(y)=\int P_{Y|X}(y|x)P_X(x)\text{dx}$ (this ...
7
votes
1answer
791 views

What's the maximum entropy probability distribution given bounds [a,b] and mean?

What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution? For example: if a=0, b=1, ...
4
votes
2answers
256 views

Maximum entropy priors in infinite dimensional spaces

Is there an extension of maximum entropy probability distributions for function spaces? For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as ...
3
votes
1answer
162 views

Ising entropy of a finite L_1 x L_2 lattice

We know the entropy per site of the 2-d Ising model from Onsager's solution. Has anybody also calculated the entropy for a finite rectangle of size L_1 x L_2 with periodic boundary conditions (i.e. on ...
5
votes
1answer
596 views

Bounding the entropy of a convolution

Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...
1
vote
0answers
54 views

Information amount of fuzzy data transfer [closed]

Suppose we have binary channel from which we are able to receive zeroes and ones. We also know apriory probability $p$ of receiving "1". Then we can calculate information amount of each digit $q$ we ...
1
vote
0answers
240 views

Incremental computation of a conditional entropy

Is it possible to compute a conditional entropy (see the two following formulas) in an incremental manner ? That is, the sets C and K are not fix: each time we have a new element c, set K may increase ...
1
vote
1answer
140 views

asymptotic behaviour of the entropy and degeneracy

For each $n \in \mathbb{N}$ let $X_n$ be a random variable taking its values in a finite set $E_n$ with $P(X_n=x_n)>0$ for all $x_n \in E_n$. Say that $X_n$ is asymptotically degenerate if ...
4
votes
1answer
395 views

Low degree polynomial approximation for the entropy function

Let $X$ be a discrete random variable with possible values $\{x_1,\ldots,x_n\}$, and let $p$ denote the probability mass function of $X$. In addition, denote $p_i=p(x_i)$. The entropy of $X$ is ...
0
votes
1answer
293 views

Robust entropy-like measure for analyzing uncertainity

I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...
1
vote
1answer
4k views

How to compute KL-divergence when PMF contains 0s?

From the Wikipedia page on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula: The way I understand this is to compute the PMFs of two given sample sets ...
8
votes
1answer
1k views

Entropy of the Ising model

Consider the standard Ising model on $[0,N]^2$ for $N$ large. By that I mean the square-lattice Ising model without external field, inside an $N$-by-$N$ square. What is its entropy for $N$ large? It ...
1
vote
3answers
2k views

Entropy of Random Signal

How would you proof that for all equal variance (Continuous) Random Signals, a Gaussian one would have the largest Entropy. Or in other words, Given a Variance Gaussian PDF maximizes the Entropy. ...