# Tagged Questions

2answers
161 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 ...
1answer
128 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, ...
0answers
176 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 ...
1answer
367 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 ...
1answer
673 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 ...
0answers
88 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 ...
1answer
94 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 ...
2answers
1k 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 ...
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 ...
3answers
440 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 $S_i^j:=\sum_{k=i}^j x_k$. ...
1answer
547 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 ...
1answer
266 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 ...
1answer
806 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, ...
2answers
260 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 ...
1answer
164 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 ...
1answer
604 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 ...
0answers
55 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 ...
0answers
241 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 ...
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 ...
1answer
404 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 ...
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 ...
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 ...
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 ...
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. ...