1
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1answer
46 views

What is the correct equation for conditional relative entropy and why

I was trying to understand the concept of conditional relative entropy. As in: $$D(P(X\mid Y) ||Q(X\mid Y))= E [\log\frac{P(X\mid Y)}{Q(X\mid Y)}]$$ I would have thought that its equations would ...
0
votes
0answers
31 views

Entropy derivation from Multiplicity

Multiplicity(W)= N!/(n1!*n2!....ni!) Entropy = 1/N * ln W = 1/N*ln N! - 1/N*sigma_for_all_i(ln ni!) As N->infinity, By Stirlings approximation ...
0
votes
1answer
54 views

calculate channel capacity and maximum conditional entropy

i want to know when it is equal channel capacity or $I(X,Y)$ maximum or where $I(X,Y)=H(X)-H(X\mid Y)=H(Y)-H(Y\mid X)$ now if we have two random variable with some specific distribution ...
1
vote
0answers
32 views

continuous, non-additive set function

I am looking for a non-additive, continuous set function from a simplex $\Pi_{n}$ of finite dimension $n-1$ into $[0,\infty]$. The motivation is as follows. Shannon defined the entropy ...
1
vote
1answer
54 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 ...
2
votes
0answers
47 views

Maximize the expected Brier Score of P relative to P.

Fix a finite set $\mathcal S$ and let $\mathcal P$ be the collection of probability functions over the Boolean closure of $\mathcal S$. Let $\beta : \mathcal P \times \mathcal S \to \mathbb R$, with ...
0
votes
3answers
80 views

Roll a dice, ignore result if result is maximum value

Let's say I have a six-faced dice, but I only want results between $1$ and $5$. One way to do that would be to roll a ten-faced dice and divide the result by two ($1-2$ becomes $1$, $3-4$ becomes ...
0
votes
2answers
47 views

Properties of joint entropy

I'm trying to show the following, but am stuck For discrete random variables $X$ and $Y$, show $$H(X,Y)\geq \max\{H(X),H(Y)\}$$ where $H$ represents entropy
1
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0answers
74 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
2answers
51 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 ...
1
vote
1answer
228 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 ...
2
votes
1answer
99 views

Equality of sets when minimizing Shannon's Entropy

Let $P = \{p_1, \ldots, p_n\}$ be a set of probabilities, i.e., $0 \leq p_i \leq 1$. $P$ is such that $\sum_{p_i \in P} p_i = 1$. I have a set of actions $\mathcal{A} = \{a_1, \ldots, a_N\}$ that can ...
0
votes
0answers
72 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
0answers
60 views

convergence of discrete random variables with finite entropy

Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ ...
0
votes
1answer
665 views

Kullback-Leibler distance between 2 probability distributions

Can I determine the Kullback-Leibler distance $$ D_{\mathrm{KL}}(P\parallel Q)=\sum_{i}\ln\left(\frac{P(i)}{Q(i)}\right) P(i) $$ between the following probability distributions? ...
1
vote
1answer
264 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 ...
1
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0answers
118 views

Incrementally compute the 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
120 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 ...
4
votes
2answers
886 views

Can I normalize KL-divergence to be $\leq 1$?

The Kullback-Leibler divergence has a strong relationship with mutual information, and mutual information has a number of normalized variants. Is there some similar, entropy-like value that I can use ...