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

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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 ...
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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 ...
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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) + ...
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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 ...
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1answer
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“Empirical” entropy.

Information entropy is usually defined as $$\text{I}_b({\bf p}) = -\sum_{\forall i}p_i\log_b(p_i)$$ i.e. the expected value of the negative logarithm of the probabilities. This is all good when we ...
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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, ...
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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%. ...
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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)] ...
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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 ...
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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 ...
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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?
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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 ...
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1answer
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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 $p>1$ ...
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1answer
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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 ...
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1answer
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Entropy Solution of $u_t+(u^2/2)_x=0$

Given the initial data $$ g(x)= \cases{ 1 & x< -1 \\ 0 & -1 < x< 0 \\ 2 & 0 < x< 1 \\ 0 & 1 < x \\ } $$ What is the entropy solution of $u_t+(u^2/2)_x=0$?
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1answer
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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 ...
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Entropy of $Y=bX$

If I have two random variables $Y$ and $U$ related as $Y=bU$, where $b>0$ is a constant and knowing that $\text{H}(x)$ represents the shannon entropy, such that: $$ \text{H}(x)=−\int \text{p}(x) ...
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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: $$H(Z)=H(X)+H(Y)$$