Questions tagged [entropy]

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37
votes
7answers
4k views

Can PRNGs be used to magically compress stuff?

This idea occurred to me as a kid learning to program and on first encountering PRNG's. I still don't know how realistic it is, but now there's stack exchange. Here's a 14 year-old's scheme for an ...
34
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6answers
7k views

Do lossless compression algorithms reduce entropy?

According to Wikipedia: Shannon's entropy measures the information contained in a message as opposed to the portion of the message that is determined (or predictable). Examples of the latter ...
30
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2answers
2k views

Simulating a probability of 1 of 2^N with less than N random bits

Say I need to simulate the following discrete distribution: $$ P(X = k) = \begin{cases} \frac{1}{2^N}, & \text{if $k = 1$} \\ 1 - \frac{1}{2^N}, & \text{if $k = 0$} \end{cases} $$ The most ...
30
votes
7answers
2k views

Is there a connection between the halting problem and thermodynamic entropy?

Alan Turing proposed a model for a machine (the Turing Machine, TM) which computes (numbers, functions, etc.) and proved the Halting Theorem. A TM is an abstract concept of a machine (or engine if ...
24
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11answers
9k views

Is von Neumann's randomness in sin quote no longer applicable?

Some chap said the following: Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin. That's always taken to mean that you can't generate ...
18
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2answers
5k views

How does an operating system create entropy for random seeds?

On Linux, the files /dev/random and /dev/urandom files are the blocking and non-blocking (respectively) sources of pseudo-random ...
17
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2answers
773 views

What's harder: Shuffling a sorted deck or sorting a shuffled one?

You have an array of $n$ distinct elements. You have access to a comparator (a black box function taking two elements $a$ and $b$ and returning true iff $a < b$) and a truly random source of bits (...
14
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3answers
2k views

Shannon Entropy of 0.922, 3 Distinct Values

Given a string of values $AAAAAAAABC$, the Shannon Entropy in log base $2$ comes to $0.922$. From what I understand, in base $2$ the Shannon Entropy rounded up is the minimum number of bits ...
10
votes
2answers
1k views

Is there a generalization of Huffman Coding to Arithmetic coding?

In trying to understand the relationships between Huffman Coding, Arithmetic Coding, and Range Coding, I began to think of the shortcomings of Huffman coding to be related to the problem of fractional ...
10
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1answer
1k views

Constrainted Optimization Problem in Matrix Entropy

I have a constrainted optimization problem in the (Shannon) matrix entropy $\mathtt{(sum(entr(eig(A))))}$. The matrix $A$ can be written as the sum of rank 1 matrices of the form $[v_i\,v_i^T]$ where $...
9
votes
1answer
2k views

How to practically measure entropy of a file?

I'm trying to measure now much non redundant (actual) information my file contains. Some call this the amount of entropy. Of course there is the standard p(x) log{p(x)}, but I think that Shannon was ...
8
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1answer
800 views

Rényi entropy at infinity or min-entropy

I'm reading a paper that refers to the limit as n goes to infinity of Rényi entropy. It defines it as ${{H}_{n}}\left( X \right)=\dfrac{1}{1-n} \log_2 \left( \sum\limits_{i=1}^{N}{p_{i}^{n}} \right)$. ...
7
votes
3answers
9k views

Shannon's entropy for an image

Shannon's entropy [plog(1/p)] for an image is a probabilistic method for comparing two pixels or a group of pixels.Suppose an image with a matrix of 3x3 has pixel intensity values ...
7
votes
1answer
676 views

Shannon Entropy to Min-Entropy

In many papers I've read that it is well known that the Shannon entropy of a random variable can be converted to min-entropy (up to small statistical distance) by taking independent copies of the ...
6
votes
4answers
5k views

Compression of Random Data is Impossible?

A few days ago this appeared on HN http://www.patrickcraig.co.uk/other/compression.htm. This refers to a challenge from 2001 - where someone was offering a prize of \$5000 for any kind of reduction to ...
6
votes
2answers
157 views

How best to statistically verify random numbers?

Lets say I have 1000 bytes that are supposedly random. I want to verify to a certain certainty that they are indeed random and evenly distributed across all byte values. Aside from calculating the ...
6
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2answers
2k views

Compressing normally distributed data

Given normally distributed integers with a mean of 0 and a standard deviation $\sigma$ around 1000, how do I compress those numbers (almost) perfectly? Given the entropy of the Gaussian distribution, ...
6
votes
1answer
936 views

Showing that the entropy of i.i.d. random variables is the sum of entropies

The shannon entropy of a random variable $Y$ (with possible outcomes $\Sigma=\{\sigma_{1},...,\sigma_{k}\}$) is given by $H(Y)=-\sum\limits_{i=1}^{k}P(Y=\sigma_{i})\;\log(P(Y=\sigma_{i}))$. For a ...
6
votes
1answer
892 views

Measuring entropy for a table (e.g., SQL results)

We're running some benchmarks for an approximative query-answering system. It's sufficient to just think of it as running some SQL queries with joins. We are counting the results returned as part of ...
5
votes
1answer
1k views

The Entropy of the phrase “Eile Mit Weile”

I want to calculate the Entropy of the phrase "Eile mit Weile". I found the probability of each letter as the following $$P(e)=\frac{4}{12}$$ $$P(i)=\frac{3}{12}$$ $$P(l)=\frac{2}{12}$$ $$P(m)=\frac{...
5
votes
2answers
318 views

How do computers compute?

This is a kind of follow-up to a question I asked on superuser, where I asked for the definitions of a 'distinghuisable state' and a 'memory cell'. My questions where properly answered, but I was ...
5
votes
1answer
742 views

Computing Von Neumann Entropy Efficiently

The Von Neumann entropy $S$ of a density matrix $\rho$ is defined to be $S(\rho)= -\text{tr}(\rho \lg \rho)$. Equivalently, $S$ is the classical entropy of the eigenvalues $\lambda_k$ treated as ...
5
votes
1answer
537 views

Mutual information intuition

I was creating an example for a casual talk on mutual information. I considered a system of two coins, which with probability 1/2 are copies of each other, and with probability 1/2 are independent. ...
5
votes
2answers
704 views

Estimate entropy, based upon observed frequency counts

Suppose I have $n$ independent observations $x_1,\dots,x_n$ from some unknown distribution over a known alphabet $\Sigma$, and I want to estimate the entropy of the distribution. I can count the ...
5
votes
1answer
2k views

Pointwise mutual information vs. Mutual information?

I am learning about information theory and mutual information. However, I am quite confused with MI(Mutual information) vs. PMI(Pointwise mutual information) especially signs of MI and PMI values. ...
4
votes
1answer
252 views

Conceptual question about entropy and information

Shannon's entropy measures the information content by means of probability. Is it the information content or the information that increases or decreases with entropy? Increase in entropy means that ...
4
votes
1answer
2k views

What units should Shannon entropy be measured in?

The only examples I've seen use bits as a measurement of entropy, but all these examples happen to use binary code alphabets. If we wanted to see how well a coding with a code alphabet of length n ...
4
votes
0answers
350 views

Is there a relationship between graph entropy and node entropy?

Eagle, et al [1] discuss the notion of node entropy and this is captured in igraph via the diversity metric. I was wondering if there was any relationship between these node entropies and the idea of ...
3
votes
1answer
1k views

Why is the Shannon entropy 0.94 in this example?

Suppose I have a decision tree in which there is a label $L$ under which is the attribute $A$ as shown below. I am given that the Shannon entropy of label $L$ is $H(L) = 0.95$. I must find the ...
3
votes
2answers
677 views

Why Shannon's Entropy is said to be a measure of information?

I got a bit of how Shannon explained to find the number of bits required to represent a message and Shannon's Entropy.But it's natural to know that to code alphabet letters you need ...
3
votes
1answer
219 views

Source entropy and other questions related to information theory

Kolmogorov-Sinai entropy (KS) explains the mathematical concept behind KS entropy. $$h ( T ) =\sup\limits_{\xi} \, h ( T , \xi )$$ defines the formula for KS where the left-hand side is nothing but ...
3
votes
1answer
191 views

Complexity of / best algorithm for finding the dichotomy that maximizes information gain?

Suppose that $X$ is a finite set with a probability measure $P$. I want to find the subset $A \subset X$ so that the information gain of conditioning on ${A, A^c}$ is maximal. That is, I want to find $...
3
votes
1answer
94 views

Dependency on adjacent blocks decreases as block count increases

The following is an excerpt from Information Theory: A Tutorial Introduction, page 65. Now, supposing the identity of each letter in English does not depend on any letter that is more than 10 ...
3
votes
1answer
90 views

Is there a fundamental difference in LIFO and FIFO entropy coding?

The top two competing entropy coders at the moment are Arithmetic Coding (AC) and Asymmetrical Number Systems (ANS). A very interesting difference between the two is that AC is FIFO and ANS is LIFO. ...
3
votes
1answer
96 views

Finding the dichotomy that maximizes information gain for a classifier?

Suppose that $\Omega$ is a finite probability space,with measure $P$ (we can take $P$ uniform). Let $C \in \{\pm 1 \}$ be a random variable on $\Omega$, the classifier. Let $$H(C) = H(C, \Omega, P) = ...
3
votes
2answers
113 views

Measuring the information of a document?

I'd like to measure how much information a document $D$ contains. Clearly, the New York Times published yesterday contains more information than my diary wrote on the same day. But, I do not know ...
3
votes
1answer
66 views

Maximum entropy probability distribution among Solomonoff priors

If we take Solomonoff's prior $m$, defined here and normalize it we get a probability mass function on all finite words. But, the pmf isn't completely determined until we fix a universal Turing ...
3
votes
0answers
126 views

How to compare conditional entropy and mutual information?

I am solving a problem of information theory. The problem reads, Consider a stationary memoryless channel specified by the channel matrix $T = \begin{pmatrix}1-q&q\\r&1-r\end{pmatrix}$. ...
2
votes
2answers
396 views

How to extract randomness from a file?

I have a generator of files with approximately 7 bits /byte entropy. The files are about 30KB each in length. I'd like to use these as sources of entropy to generate random numbers. Theoretically I ...
2
votes
1answer
169 views

Increasing entropy of random walk

Let $P$ be a transition matrix of a random walk in an undirected (may not regular) graph $G$. Let $\pi$ be a distribution on $V(G)$. The Shannon entropy of $\pi$ is defined by $$H(\pi)=-\sum_{v \in ...
2
votes
1answer
99 views

Entropy notation: What does this mean?

If you look at page 13 of the lecture slides here there is this line $H(Y) = H((1-\pi)(1-\alpha), \alpha, \pi(1-\alpha))$ I don't really understand what the term on right hand side is. At first I ...
2
votes
1answer
75 views

How to prove Landauer's principle [closed]

I have some questions about energy emitted when one bit of information is processed. Landauer's principle states the minimum possible amount of energy required to erase one bit of information is <...
2
votes
1answer
150 views

How realistic is the i.i.d assumption in the definition of Shannon's entropy?

Let me first say I come from a physics background and have about zero exposure to computer science, so the question may be very naive. Shannon's entropy looks perfectly natural and useful from a ...
2
votes
1answer
108 views

Kraft's inequality and Shannon's noiseless coding theorem for an encoding

A discrete memoryless source W has words $w_1,w_2,w_3,w_4,w_5,w_6$ that occur with probablilities $0.05,0.05,0.15,0.2,0.25,0.3$ respectivley. Does there exist a compact instantaneous binary encoding ...
2
votes
1answer
35 views

Is there an algorithm to achieve optimal compression in a “streamed” manner, assuming equal probability of each possibility?

(Sorry for the question title; edits are welcome.) Let's say that you have a set of data made of repeating units, consisting of a value with $2$ possibilities, a value with $3$ possibilities, $5$ ...
2
votes
1answer
148 views

Can Von Neumann bit extraction be made more efficient?

I want to develop a previous question regarding Von Neumann debiasing /randomness extraction. The typical solution (as posted) is to take pairs of throws and output a bit based on a comparison of ...
2
votes
1answer
116 views

Extracting Randomness from Mouse Acceleration

I'm working on trying to make an "entropy pool" that will be fed as input into an RNG (as in, ex, Fortuna). In order to do so, I need to take various collected data and extract as much entropy as ...
2
votes
1answer
22 views

Proof high-entropy sequences are hard to compress

When comparing random-variable sequences generated from probability distributions, what's a formal proof that a distribution with higher entropy produces a sequence that's "harder" to compress? In ...
2
votes
1answer
58 views

When Huffman coding is inefficient?

I have a question regarding the redundancy of Huffman coding. I know that for a general prefix code we have the following inequality: $$ H(X) \le R \le H(X) + 1 $$ $R$ being the rate (average ...
2
votes
1answer
40 views

Using all the entropy in an biased bit

Suppose we have $n$ bits of random-looking data, and we want to encode it in such a way that instead of 1/2 the bits being 1's, we have (say) 3/4 the bits being 1's. The entropy of each bit in the new ...