# Questions tagged [entropy]

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### Calculate joint entropy of a Hamming Code over a Binary Symmetric Channel?

I have a normal (7,4,3) Hamming Code over GF(2) and a parity check matrix for it (not posted, because I don't think it's involved). I have a set X of 4 bit source vectors called x. They all have ...
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 1 answer 257 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 1 answer 190 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 1 answer 170 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 0 answers 82 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 1 answer 125 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 1 answer 90 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 2 answers 218 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 2 answers 230 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 0 answers 121 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 2 answers 54 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 0 answers 120 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 1 answer 2k 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 2 answers 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 0 answers 51 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 1 answer 89 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 2 answers 589 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 1 answer 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 2 answers 911 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 ... 13 votes 4 answers 8k 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 1 answer 131 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 1 answer 45 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 1 answer 200 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? 6 votes 1 answer 1k 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 0 answers 82 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= ... 21 votes 2 answers 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 1 answer 268 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 ... 5 votes 1 answer 403 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 0 answers 143 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 7 answers 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 7 answers 5k 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 1 answer 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 ... 7 votes 2 answers 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 0 answers 375 views ### Is there a relationship between graph entropy and node entropy? Eagle, et al  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 2 answers 849 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 2 answers 543 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 1 answer 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 1 answer 273 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 2 answers 960 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 1 answer 739 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 1 answer 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 1 answer 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 1 answer 197 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 ...
I'm reading a paper that refers to the limit as n goes to infinity of Rényi entropy. It defines it as ${{H}_{n}}\left( X \right)=\dfrac{1}{1-n} \log_2 \left( \sum\limits_{i=1}^{N}{p_{i}^{n}} \right)$. ...