# Questions tagged [entropy]

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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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### Is von Neumann's randomness in sin quote no longer applicable?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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 ...
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### 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 ${{H}_{n}}\left( X \right)=\dfrac{1}{1-n} \log_2 \left( \sum\limits_{i=1}^{N}{p_{i}^{n}} \right)$. ...
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 ...
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### 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 ...
162 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 ...
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### 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, ...
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### 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 ...
932 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 ...
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### 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 ...
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### 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 ...
129 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 $T = \begin{pmatrix}1-q&q\\r&1-r\end{pmatrix}$. ...