Questions tagged [information-theory]

Questions about Information theory, entropy, and information content of various sources

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Example of channel where capacity is achieved without a uniform distribution on the output alphabet

The capacity of a discrete memoryless channel is given by the maximum of the mutual information over all possible input probability distributions. That is \begin{align} C &= \max_{p_X} I(X:Y) \\ &...
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36 views

Is a complete code always optimal?

According to wikipedia, Kraft's inequality holds with equality when a code is complete. Huffman encoding produces a complete code that is optimal. Are all complete codes optimal and vice versa?
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Information Theory: Comparing surprisal of words with varying count frequency

This is a very broad question, I'm not sure if cstheory is the better place. How can I compare the conditional surprisal of words that vary in frequency? $S(w|context)=−log(p(w|context))=−log(\frac{...
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44 views

information theory, find entropy given Markov chain

There is an information source on the information source alphabet $A = \{a, b, c\}$ represented by the state transition diagram below: a) The random variable representing the $i$-th output from this ...
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1answer
19 views

Is the capacity achieving input of a discrete memoryless channel unique?

Consider a classical discrete memoryless channel (DMC). Let $p$ be an input probability distribution and $Q$ be the channel's transition matrix. $q = Qp$ is a valid output probability distribution. ...
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Do all Cellular Automata have some kind of information boundary? Can all Cellular Automata be modelled with the Bekenstein Bound?

Since they are discrete models, do they have some kind of information boundary? Can all Cellular Automata models be related to the Bekenstein Bound? https://en.wikipedia.org/wiki/Bekenstein_bound
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1answer
28 views

Examples of exact computation of Kolmogorov complexity?

First question: It is known that Kolmogorov Complexity (KC) is not computable (systematically). I would like to know if there are any "real-world" examples-applications where the KC has been computed ...
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1answer
21 views

Prove that the upper bound in the Noiseless-coding theorem is strict

Given a probability distribution $p$ across an alphabet, we define redundancy as: Expected Length of codewords - entropy of p = $\ E(S) - h(p)$ Prove that for each $\epsilon$ with $0 \le \epsilon \...
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31 views

Reaching Shannon capacity of a channel

Suppose I have the following from alphabet $\mathcal{X} = \{0 ,1\}$ to $\mathcal{Y} = \{0 ,1\}$. The channel simply does \begin{align} 0 \rightarrow 0&\quad \text{with probability 1} \\ 1 \...
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35 views

Is entropy a good indicator of the quality of a lossy compression?

Say I want to quantitively evaluate the effectiveness of several color-to-grayscale conversion algorithms, which can be considered as lossy compression. Would entropy be a good indicator? To ...
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43 views

Coding for data compression with large target's symbol set (where the target symbol set is larger than the source symbol set)

For data compression, every codding that I've seen is binary. It means we convert a language with $N$ symbol size to a language with $M=2$ symbol size. For example, in Huffman coding, the goal is to ...
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go back n protocol scenario consequences

i am not sure with tihs question. i will post it here along with my attempt: the following scenario is regarding go back protocol n, with 3 bits for sequential numbering, assuming that the following ...
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1answer
29 views

Given two data feeds, find out if they capture the same information

Say, there are two camera feeds, how can I establish if they were filming the same scene? It seems plausible that there are algorithms that somehow calculate mutual information and detect "causality ...
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1answer
36 views

How to calculate information gain in ID3?

I am trying to implement a decision tree classifier using ID3 algorithm. According to Aritificial Intelligence - A Modern Approach, information gain of attribute A ...
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55 views

finding the overhead and distance of an unknown code based on message making algorithm

for an information word M with m bits that is coded as following: M is coded into a word A using an unknown code that allows detection of not more than one error. the code word is the word obtained ...
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52 views

Is it possible to achieve greater than perfect compression using machine learning and big data?

Imagine Google wanted to make their chrome browser faster. Let "database" be all the machines which serve content from Google's servers, including Search and Google cloud services. Google begins using ...
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743 views

Under what conditions does the function C = f(A,B) satisfy H(C|A) = H(B)?

Suppose we have a function $f$, $$ C = f(A,B), $$ where $A$, $B$ and $C$ are random variables. I notice that when the random variables are binary ($\{0, 1\}$) and $f$ is the XOR operation, we have ...
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65 views

What doest it mean: “computer is an intelligence amplifier”?

There is one example in Kolmogorov complexity books and related articles: Consider we have a monkey at a typewriter and a monkey at a computer keyboard. If the monkey types at random on a typewriter,...
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28 views

Sphere packing inequality for error-correcting codes

i am wondering if the following inequality is correct: if a code allows repairing of no more than k errors (inclusive, included) and m is the number of information bits and r the check bits, then $$∑^...
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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 ...
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Prove that $I(A;B|C)=0$ given $I(A;B)=0$

Let $A$, $B$ and $C$ be 3 discrete random variables. If $A$ and $B$ are independent ($I(A;B) = 0$, where $I$ represents the mutual information), how can we prove that $I(A;B|C)=0$? When I draw a Venn ...
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Does a binary code with length 6, size 32 and distance 2 exist?

The problem is to prove or disprove the existence of $C$, s.t., $|c| = 6,\forall c\in C$; $|C| = 32$; $d(c_i,c_j)\geq2,1\leq i<j\leq32$. ($d$ stands for hamming distance) I tried to construct a ...
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Prove the MinAveCodeLen of a product information source is less than the sum of that of the multiplicand and multiplier source?

The product of 2 independent sources $(S_A,P_A)$ and $(S_B,P_B)$ is defined as $$ (S,P)\text{ s.t. }S = \{s_As_B|s_A\in S_A,s_B\in B\}\text{ and }\ P(s_As_B) = P_A(s_A)\cdot P_B(s_B)\,\forall s_A\in ...
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How to show that in noiseless coding theorem, the bound $\mathrm{MinACL}<H(P)+1$ is tight?

The theorem states that $$ H(P)\leq\mathrm{MinACL}(P)<H(P)+1 $$ where, $\mathrm{MinACL}$ means the minimum average code word length of a given information source, i.e. the average code word ...
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217 views

Huffman Coding vs Arithmetic Coding

Are there conditions under which given text $T$ that huffman code and arithmetic code will produce the exact encoding of $T$?
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2answers
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What is the term for two file formats that describe the same amount of information?

I am working on file formats conversion. Some of them can be converted back and forth to others without losing any information (fields or precision of the numbers encoded), others file format do not ...
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What is the compressibility of this simple “book”?

Compressibility is defined as $$C=\frac{2^{HN}}{2^{H_{max}N}}$$ The book is made up of a simple alphabet of only {a,b,c,d} which occur with probabilities $$P(a)=0.2, P(b)=0.4, P(c)=0.1, P(d)=0.3$$ ...
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60 views

Why it is not a Huffman code

I have been given several examples I the aim is to explain why it is not a Huffman code. So, for instance, the first one was: $\{00,01,10,110\}$ This code is not Huffman becuase it has just one ...
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Finding the 'capacity' of a network by treating it as a channel

Let's say I have a fairly basic network with two senders and two receivers. I'm wondering if I can treat this network as an effective communication channel that takes an input alphabet made up of the ...
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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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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 ...
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I think you can always compress compressed data, is it true?

In compressed data, repetition of same pattern is not a lot, so, you can expect it to be with space inside to contain always. I found a way to compress data without limitation. Is it right? Am I ...
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Computing/sketching essential bit content of a binary source

I'm given the Bernoulli distribution of a biased coin toss with probability distribution $P_X = \{0.2, 0.8\}$ over the alphabet $\mathcal{A}_X=\{0,1\}$. I want to sketch the normalized essential bit ...
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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 ...
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How many prefix code we can find for a given distribution of probability?

A source emits 5 signals s1, s2, s3, s4 and s5 whose probabilities are as follows: 1/3, 1/3, 1/9, 1/9, 1/9, 1/9. How many prefix codes we can construct on A={a, b, c}? And how many are there with the ...
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21 views

What is the relation between the quantity of information of $A_i$ and $A$, where $A=\bigcup_iA_i$?

Let $A$ be an event divided into 4 events $A_i$ with the same probability. Why does the quantity of information of $A_i$ satisfy $$ I(A_i) = I(A) + \log (4)?$$
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How to use HITS algorithm to rank pages

I am trying to implement a simple web search engine mechanism with using HITS algorithm. I understand how the algorithm work and produce hubs and authority values for each page. But how can I use this ...
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69 views

fully homomorphic encryption with information-theoretic security?

An encryption algorithm with information-theoretic security is one which even with infinite amount of computation cannot be broken. That is, given only the ciphertext, no amount of computation can ...
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47 views

what does the redundancy of a code means?

I was reading a paper about a transposition and single deletion error correcting code and they claim that the redundancy of the code was only $\log(6n-3)$ bits. But what does that means? I was ...
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77 views

Infinite Huffman Tree

We have to derive an optimal binary encoding for the infinite set of symbols $\{s_1, s_2, \dots \}$. They're distribution is given by $$p(s_i) = 9 \cdot 10^{-i}$$ My intuition was to use a Huffman ...
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25 views

Classify/Distinguish between 8008 binary grids, with 13 queries

I have $8008$ binary grids of size $6 \times 10$ (they are all grids with the property described below), which I want to distinguish between with at most $13$ queries. A query will determine if the ...
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1answer
41 views

Maximizing entropy under constraint

How do I prove that entropy is maximal for $P(A_2) = \cdots = P(A_n) = (1-a) /(n-1)$ while $P(A_1) = a$ (a fixed number) and $A_1,…, A_n$ is a partition of the sample space?
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How to know a certain grammar is parse-able

Is it possible to parse all kinds of structured data and give them a semantic meaning? For example, C++ is a really complicated language and I could never imagine a parser would be possible for it. ...
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Can data be compressed to size smaller than Shannon data compression limit?

I was reading about data compression algorithms and the theoretical limit for data compression. Recently I encountered a compression method called "Combinatorial Entropy Encoding", the main idea of ...
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Finding the total capacity of two communication channels

I have the transition matrices of two communication channels. I am able to find the capacity of each by performing an optimization calculation, however I need the total capacity of the two channels. ...
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34 views

How hexadecimal representation is more compact and intelligible for documentation?

My textbook says, "Instead, it is far better to use a hexadecimal representation for documentation purposes. Whether or not a code represents a binary number, it can be treated as ...
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30 views

Does data entropy depend on the arrangement of the characters in a file?

From what I understand data entropy controls the limit of data compression and it depends on the probability of the characters in the file. Assuming that we have a file of size 256 bytes containing ...
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1answer
80 views

A necessary [and sufficient?] condition for the number of comparisons required to sort $n$ elements

I am familiar with the decision tree based argument for the minimal number of comparisons required to sort $n$ distinct elements - Since there are $n!$ permutations on the $n$ elements, the decision ...
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1answer
70 views

Proof of Lower Bound for Deterministic Distinct Elements Algorithm

There is a proof in this document (page 8, Section 4, Lemma 3: https://inst.eecs.berkeley.edu/~cs170/fa16/lecture-11-29.pdf) that mirrors a proof my professor gave in my algorithms class. The lemma ...
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96 views

Applying Shannon Entropy to data storage

I know Shannon Entropy is defined for messages. When it's said the size of a file stored in a HDD is 1 MB, are we talking about the Shannon Entropy? If so, how do we extend the definition to static ...