# Questions tagged [shannon-entropy]

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130 views

### Does this notion of entropy have a name?

Recently I stumbled upon the following notion of entropy which seems quite natural to me. I am looking for its "real" name and/or any references where it might come up. I tried searching ...
153 views

### Error in entropy properties in Mathematical Theory of Cryptography by Claude E. Shannon

I am reading this classic paper by Claude E. Shannon and I think there may be a couple of errors in his description of the properties of Entropy/Uncertainty. The screenshot shown at the bottom of this ...
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### Why isn’t information-probability relationship linear? [closed]

I am completely new to information theory. I was learning about information content but couldn’t make sense of why the relationship between information content and probability isn’t linear? And why it ...
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### Capacity of spike-based neuronal code

Assume that a neuronal population $A$ is connected to a neuronal population $B$ by a bunch of synapses - one-directional channels that propagate spikes. For simplicity assume that the current ...
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### Generating $k$ random bits from a pdf with entropy $H(p) = k$

All the sources online say that, intuitively, a distribution with entropy $k$ has $k$ bits of pure randomness in it. So can we formalize this as follows? Suppose I can only sample from my distribution,...
248 views

### Entropy-like quantity

For $p\in[0,1]^{\mathbb{N}}$ and $\alpha\ge1$, define $$H_\alpha(p) = \sum_{i\in\mathbb{N}}p_i|\log(p_i)|^\alpha.$$ When $\sum_i p_i=1$ and $\alpha=1$, $H_1(p)$ is just the Shannon entropy of the ...
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### Can entropicly secure encryption algorithms be used on low-entropy messages by adding noise

There exist information-theoretic notions of security like Shannon's "perfect security" that one-time pads exhibit. All methods which achieve perfect security will require long keys, however. If we ...
558 views

### Relation between variance and mutual information

Given two discrete random variables $X,Y$ such that $X,Y \in \mathbb{R}$ and $0 \leq X,Y \leq 1$, is it true that $$|\text{Cov}[X,Y] \leq \sqrt{\frac{1}{2} \text{I}[X,Y]}|.$$ This bound may be ...
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### Word length using entropy : Maximum entropy criteria

The question is based on research paper titled, Markovian language model of the DNA and its information content In the supplementary document, the Authors show how they determine the word length of ...
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### Lower bound on the number of objects in the universe [closed]

From Cover & Thomas' Elements of Information Theory: Player A chooses some object in the universe, and player B attempts to identify the object with a series of yes–no questions. Suppose ...
810 views

### How does the Multiplicative Weights Update method maximize entropy?

"The Multiplicative Weights Update (MWU) method is known to maximize both utility and entropy". This is a comment by C. Papadimitriou on MWU. I understand that MWU maximizes utility as it solves ...
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### Inf-entropy rate and min-entropy

I am reading the paper "Generating random bits from an arbitrary source: fundamental limits" by Vembu and Verdu. This paper is written in the language of information theory, however, I need to ...
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### What is entropy of a variable described by Knightian uncertainty?

Given a discrete variable whose value is characterized by Knightian uncertainty, that is, belief and plausibility, as in Dempster-Shafer theory, what is its entropy?
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### Entropy criterion of efficiency for (comparison using hashing)

I understand that hash is effective iff the "domain" size is smaller than the size of the "general set" - set of all possible objects. E.g., "domain" is the set of valid english phrases with length ...
153 views

### How hard is it to compute an approximately optimal non-greedy CART tree?

The question itself is closer to the bottom of this post, and is formulated without any rerefence to the term "CART". Motivation: In traditional CART (Classification and Regression Trees), one ...
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### What is full-entropy bit-strings?

I was going through the description of NIST Randomness Beacon. I would like to know the meaning of the term full-entropy bit-strings used in the third paragraph.
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### Geometric Intuition behind Locally testable codes

Conventional coding theory provides a good geometric picture behind linear error correction codes in terms of Hamming distance. What additional geometric requirement one should add to make a code ...
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### Are Shannon entropy and Boltzmann entropy mutually convertible?

Are Shannon entropy and Boltzmann entropy mutually convertible, much like mass and energy according to Einstein's formula?
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### On the entropy of a sum

I am looking for a bound on the entropy $H(X+Y)$ of the sum of two independent discrete random variables $X$ and $Y$. Naturally, $$H(X+Y) \leq H(X) + H(Y) ~~~~~~(*)$$ However, applied to the sum of $n$...
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### Information channel with symmetric channel matrix

It took me a while to figure out that a "symmetric channel" does not mean a channel with a symmetric channel matrix. (Rather, "symmetric channel" means that the rows of the matrix are all permutations ...
736 views

### Arithmetic coding, the termination symbol, and the empty string

Suppose the source alphabet is $a, b, c$ with $a$ as the termination symbol and so the unit interval is correspondingly divided as $[0, P(a), P(a)+P(b), 1]$. Strings consisting of a bunch of $b$'s ...
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### Landauer's principle internals - how it works

I attached a picture, where the energy dissipation (entropy increase) on information erasure is explained. Is the explanation correct? "RESTORE TO ONE" - is it correct to identify the operation as "...