Questions tagged [entropy]
The entropy tag has no usage guidance.
97
questions
3
votes
1answer
237 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 $...
2
votes
1answer
176 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
153 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 ...
1
vote
0answers
75 views
Approximate conditional entropy
Given a set of random variables $X = \{x_1, x_2, \dots, x_n\}$. If the conditional entropy for all $Y \subset X - \{X_i\}$ where $|Y| \leq 5$. How to approximate conditional entropy when $|Y| = 10$ ...
3
votes
1answer
120 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 ...
0
votes
1answer
85 views
Numerical example of theoretical diff file size using Kullback Leibler?
I understand that the theoretical size of a diff patch between two similar files can be calculated using Kullback Leibler (KL) as described @ Wikipedia. Can anyone point me to a numerical example of ...
7
votes
2answers
196 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 ...
4
votes
2answers
163 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 ...
1
vote
0answers
116 views
Can file entropy be predictable?
I am working on some lossless compression topic and looking on any resources or prior studies on lossless file entropy predictability.
Assuming we know type of the file (we can tell its belongs to ...
1
vote
2answers
42 views
Amount of information in scaled-down images
Does an image that is scaled down lose more Information when
calculating averages of pixels rather than selecting single pixels?
One way to scale down an image is to replace 2x2 pixel blocks with ...
1
vote
0answers
108 views
Find minimum conditional entropy
Task : Given $X$ random variables. Find out the minimum conditional entropy for a variable $x_i \in X$ when $x_i$ is conditioned upon any combination $k$ remaining variables.
Find $min(Entropy (x_i | ...
5
votes
1answer
1k 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 ...
12
votes
2answers
2k 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 ...
1
vote
0answers
49 views
Approach to data compression with unknown patterns [closed]
Let's say I have a file with symbols of an alphabet [0,N-1], N around 50. The actual data is a file with a lot of lines, each containing 100 of these symbols.
I want to compress this data. By doing a ...
3
votes
1answer
74 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 ...
2
votes
2answers
525 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 ...
10
votes
1answer
3k 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 ...
3
votes
2answers
833 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 ...
11
votes
4answers
7k 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 ...
2
votes
1answer
119 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
41 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 ...
2
votes
1answer
188 views
Is a very long plain text password harder to crack than a short complicated password? [closed]
Is it true that a password consisting of the alphabet, even of common known names is much harder to find for a computer program than a short password, even though it uses numbers and other characters?
5
votes
1answer
936 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.
...
0
votes
0answers
78 views
Entropy of residuals and noise
Relation of Entropy and SNR : Based on this question and answer, I had another question that struck me and I am curious to know, if somebody can shed some light, on the following situation: $y= ...
20
votes
2answers
6k 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 ...
2
votes
1answer
243 views
Relationship between message entropy and complexity of the best algorithm
Is it possible to estimate number of steps in best possible algorithm for classification of messages, using entropy of messages?
E.g. linear search problem. We have an ordered set of incomparable ...
4
votes
1answer
360 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 ...
3
votes
0answers
142 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
$T = \begin{pmatrix}1-q&q\\r&1-r\end{pmatrix}$. ...
32
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 ...
40
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 ...
6
votes
1answer
1k 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
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, ...
4
votes
0answers
369 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 ...
5
votes
2answers
808 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
2answers
452 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 ...
10
votes
1answer
2k 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 $...
0
votes
1answer
261 views
Notions of information content and randomness of binary square matrix
We have well established theory for measuring the information content and randomness of binary strings. Notions such as Shanon entropy and Kolmogorov-complexity were developed for binary strings.
For ...
18
votes
2answers
907 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 (...
7
votes
1answer
723 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 ...
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. ...
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 ...
2
votes
1answer
190 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
$$H(\pi)=-\sum_{v \in ...
8
votes
1answer
946 views
Rényi entropy at infinity or min-entropy
I'm reading a paper that refers to the limit as n goes to infinity of Rényi entropy. It defines it as ${{H}_{n}}\left( X \right)=\dfrac{1}{1-n} \log_2 \left( \sum\limits_{i=1}^{N}{p_{i}^{n}} \right)$. ...
4
votes
1answer
2k views
What units should Shannon entropy be measured in?
The only examples I've seen use bits as a measurement of entropy, but all these examples happen to use binary code alphabets. If we wanted to see how well a coding with a code alphabet of length n ...
7
votes
3answers
10k views
Shannon's entropy for an image
Shannon's entropy [plog(1/p)] for an image is a probabilistic method for comparing two pixels or a group of pixels.Suppose an image with a matrix of 3x3 has pixel intensity values
...
6
votes
1answer
1k views
Measuring entropy for a table (e.g., SQL results)
We're running some benchmarks for an approximative query-answering system. It's sufficient to just think of it as running some SQL queries with joins. We are counting the results returned as part of ...
3
votes
1answer
227 views
Source entropy and other questions related to information theory
Kolmogorov-Sinai entropy (KS) explains the mathematical concept behind KS entropy.
$$h ( T ) =\sup\limits_{\xi} \, h ( T , \xi )$$
defines the formula for KS where the left-hand side is nothing but ...
