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

learn more… | top users | synonyms

0
votes
1answer
480 views

equivalence between uniform and normal distribution

The principle of insufficient reason says that all outcomes are equiprobable when we have no knowledge to guess otherwise. I understand this and that this corresponds to uniform distribution. However, ...
6
votes
0answers
80 views

Approximation of Shannon entropy by trigonometric functions

Define Shannon entropy by $$I(p) = -p \log_2 p$$ Numerical experimentation shows that $\sin(\pi p)^{1-1/e}$ is a good approximation to $I(p) + I(1-p)$ on $[0,1],$ never differing by more than 3.3%. ...
7
votes
1answer
1k views

Entropy of a binomial distribution

How do we get the functional form for the entropy of a binomial distribution? Do we use Stirling's approximation? According to Wikipedia, the entropy is $\frac1 2 \log_2 \big( 2\pi e\, np(1-p) \big) + ...
3
votes
1answer
140 views

Question about entropy

Let $(X,A,\nu)$ be a probability space and $T\colon X\to X$ a measure preserving transformation $\nu$. Take a measurable partition $P=\{P_0,\dots,P_{k-1}\}$. Let$I$ be a set of all possible ...
3
votes
2answers
902 views

Derivation of the maximum entropy distribution

I am reading a book and having trouble following something. The problem is to try to maximize the differential entropy $-\int_{0}^{\infty}p(r)\log p(r)$ under the constraints that ...
1
vote
0answers
188 views

How can I determine the upper limit of Shannon Entropy?

I know that the maximum possible Shannon Entropy for an alphabet $X$ is $\log|X|$, where Shannon Entropy is: $$H(X) = - \sum_{x \in X} \; p(x) \log p(x)$$ but how is this upper limit computed?
7
votes
2answers
248 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 ...
3
votes
1answer
36 views

Do the pth powers of $p$-norms define the same partial ordering on the set of all probability distributions for all $p>1$?

Consider the $p$-th power of the Schatten $p$-norm $||q||_p$ of a probability distribution $q$ , ie, the function $\sum_j q_j^p$, where $\sum_j q_j = 1$ and $q_j \geq 0$. For fixed $q$ and $p>1$ ...
3
votes
1answer
117 views

convergence of entropy and sigma-fields

This question is related to this one. Let $(X_1, X_2, \ldots)$ be a sequence of random variables such that each $X_n$ takes its values in a finite space, say $\{0,1\}$, and the $\sigma$-field ...
0
votes
2answers
988 views

Calculating conditional entropy given two random variables

I have been reading a bit about conditional entropy, joint entropy, etc but I found this: $H(X|Y,Z)$ which seems to imply the entropy associated to $X$ given $Y$ and $Z$ (although I'm not sure how to ...