Questions tagged [information-theory]

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

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Is arithmetic coding slightly more efficient than rANS?

I'm extending a framework for lossy compression of multidimensional floating-point data. At some point in the pipeline, sequences of symbols from a non-uniform distribution are losslessly compressed ...
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1answer
20 views

Probability of loss using a binary symmetric channel

Today we talked about Information Theory and the binary symetric channel. For newbies here is a little explanation : For instance if I want to send a binary to someone : The bit will be "flipped&...
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1answer
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AEP with a Twist!

We know by AEP that if random variables $X_1,X_2,...$ are i.i.d. drawn from $P_X$ then the probability of the vectors in the weak typical set $$A_{\epsilon}^n = \{\vec x \in \mathcal{X}^n: |\frac{-1}{...
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1answer
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Intuitive explanation on why stochastic encoding performs better in channel coding

I am a little confused about stochastic encoding in channel coding. For example, in the identification problem (R. Ahlswede and G. Dueck, “Identification via channels”), the authors claim that we can ...
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18 views

Capacity of broadcast channels

It is given in the book by El Gamal that The capacity region of the DM-BC depends on the channel conditional pmf $p(y_1 , y _2 |x)$ only through the conditional marginal pmfs $p(y_1 |x)$ and $p(y_ 2 |...
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30 views

Proving an entropy inequality

I am given that $Z$ is independent of $(X,U)$, where $Z$ and $X$ are binary random variables while $U$ is an arbitrary random variable. I need to prove the following: $$ H(X\oplus Z|U) \geq H(X|U)$$ ...
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12 views

Joint typicality and distance between the vectors

In the book by Cover and Thomas,the author says that We first review the single-user Gaussian channel studied in Chapter 9. P Here Y = X + Z. Choose a rate R < 12 log(1 + N ). Fix a good ($2^{nR}$ ...
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1answer
22 views

How is expected value in entropy derived?

I was self learning about entropy and came across this equation. $$ H = - \sum p(x) \log p(x) $$ The equation for entropy in expected value, $$ H(x) = \operatorname*{\mathbb{E}}_{X \sim P}[I(x)] = -\...
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1answer
42 views

Why is channel capacity of AWGN infinite?

My professor taught us that channel capacity of AWGN channel is infinite without any input power constraints. The noise is $Z \sim \mathcal{N}(0,\sigma^2) $. There is no constraint on input signal. I ...
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Significance of model in arithmetic coding

I am trying to understand the concept of arithmetic coding, i understand how the range is subdivided after each character is read from the string. But i am unable to understand why using a more ...
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Find the number using binary search against one possible lie

We all know this classic problem, "there is some hidden number and you have to interactively guess it.", which could be solved using binary search when we know that maximum number that we can guess. ...
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1answer
21 views

Worked out example of Slepian-Wolf Theorem

Note: First posted this on Theoretical Computer Science Stack Exchange, but deleted it from there since it seems to be off-topic. The Slepian-Wolf theorem states that sequences of outputs from two ...
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3answers
139 views

What is implied probability, in the context of universal codes?

From Wikipedia: Each universal code, like each other self-delimiting (prefix) binary code, has its own "implied probability distribution" given by $ p(i) ={2}^{-\ell(i)} $ where $\ell(i)$ is ...
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1answer
41 views

Sums of $2^{-l}$ that add to 1

Consider the following problem: You are given a finite set of numbers $(l_k)_{k\in \{ 1, ..., n \}}$ such that $\sum_{k=1}^n2^{-l_k}<1$. Describe an algorithm to find a set $(l'_k)_{k\in \{ 1, .....
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Capacity of channels: Why do we need Blahut-Arimoto algorithms?

The capacity of a noisy channel $\mathcal{E}_{X\rightarrow Y}$, where the channel is given as a conditional probability distribution $p(y|x)$, is $$C = \max_{p_X}I(X:Y),$$ where $I(X:Y)$ is the ...
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20 views

Complexity of maximization of entropy of Hamming distance of bitstrings

We have a set of possible "key"s $S$ represented by bitstrings of length $k$. In other words, $S$ contains an arbitrary subset of all bitstrings of length $k$. For example, when $k=3$, it can be $S = \...
2
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1answer
26 views

on-the-fly decompress a flat-file database

I'm facing the following problem. I have a flat-file database (e.g. CSV). Since it's relatively large to store in memory, I'd like to compress it. Given a key, I need to return the uncompressed text (...
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1answer
34 views

Equivalence of two definitions of mutual information

I am learning quantum computing and as a background study, I am currently learning fundamentals of classical information theory. I thought it best to ask my doubts here. In Nielsen and Chuang, it is ...
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2answers
64 views

Data Compression :Compress a Compressed File

Suppose we have file A that has been compressed by the the method B and the output-file is C, now if I am not wrong We can not compress C more by method B, but there might another method=algorithm D ...
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1answer
50 views

How a Data Compression Software Reads a File as pure Binary File and makes Output?

I have an hybrid compression technique I want to implement, my implementation is (so far): I can encode a string into a encoded compressed string. These are binary strings. For example, I read texts ...
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1answer
29 views

Encoding System that Assign Same Number of Bits for Each Character

I am trying to get a binary string that has been converted from text of a text file, I am able to get that but the problem is, I need each character to be represented by same number of bits, but that ...
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1answer
36 views

What is the difference between rateless and online encoding?

Definitions of Rateless encoding and Online encoding are as follows. Error-correcting codes that employ no fixed block length are called rateless or fountain codes. Online encoding refers to the ...
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24 views

Auxiliary random variables in the analysis of the private information of wiretap channels

I am following Section 13.2 of Mark Wilde's book. I reproduce the question here for completeness. Consider a wiretap channel $X\rightarrow Y,Z$ defined by the conditional probabilities $p(y,z|x)$ for ...
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45 views

Decomposition of Mutual Information

I came across a book where the author uses the following property of mutual information: Let $X$,$Y$,$Z$ be arbitrary discrete random variables and let $W$ be an indicator random variable. $$ (1)\ \ ...
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1answer
144 views

Collision entropy definition

The collision entropy is defined as the Renyi entropy for the case $\alpha = 2$. It is given by $$\mathrm{H}_{2}(X)=-\log \sum_{i=1}^{n} p_{i}^{2} \tag{1}$$ Take two random variables $X$ and $X'$ ...
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1answer
55 views

Information theory of instruction set architecture design?

Information theory to a large extent deals with how to efficiently encode messages given a probability distribution over messages. Intuitively, it seems like we can think of machine instructions (or ...
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1answer
24 views

Example of a prefix-free code

I came across the following question: A source $X$ emits symbols from the alphabet $A_x$ with $|A_x| = 8$. We want to construct a prefix-free source code for this source. We want to find a code with ...
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1answer
28 views

Capacity of a discrete memoryless channel

For an integer $I$, the input-output relationship of a discrete memoryless channel is given by: $Y = X + Z$ (mod $I$, i.e. sum indicates a modular addition) where $I ≥ 2$, and • $X$ is an integer ...
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17 views

Is there a word for the fact that all data representations are equal?

For programming languages, we have the concept of Turing completeness which expresses the fact that all computers and all languages are equal in their ability to represent any algorithm so long as we ...
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1answer
35 views

Upper bound for set disjointness under product distributions

For the set disjointness problem in the 2-party model of communication complexity, Alice is given an input $X$ and Bob is given input $Y$, $X$ and $Y$ are $n$-length bitstrings (sampled from some ...
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51 views

combination network

Can any one help me find a reference about how to describe the connectivity between the nodes (H relays) and users in combination network ? I mean how exactly I can draw the connection for any number ...
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53 views

Is arithmetic coding restricted to powers of $2$ in denominator equivalent to Huffman coding?

With restriction to $\frac{k}{2^n}$ as line segment ends, does arithmetic coding degrade to Huffman coding? As far as I can tell, each symbol will be encoded with an integer amount of bits, which is ...
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2answers
102 views

In information theory, why is the entropy measured in units of bits?

In information theory, we have the quantity "information". Suppose we have some discrete random variable $X$, that can take values $\{{a,b,c\}}$ with corresponding probability distribution $\{{\frac{...
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37 views

Fundamental motivation behind the use of bits and binary representation

This is a naive question, but what makes binary representation special from a theoretical standpoint and from the standpoint of information theory? If for technical reasons building ternary computers ...
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1answer
51 views

Standard information-theoretic lower bound?

There should be a simple argument, but I'm struggling to see it. Suppose Alice has a string $x \in \{0, 1\}^n$ and sends a message $s = s(x)$ to Bob. And suppose that given $s$, Bob can reconstruct ...
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1answer
35 views

Parity vs Parity bit vs Parity sum

Assume that $B_b$ denotes the finite set of bitstrings of length $b$, if we are given its subset $A = \{e_i\}, i \in \{0,... n\}$, such that $e_i \in B_b$, what is "the parity sum of As bitstrings" ($...
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1answer
58 views

Understanding simulated annealing information theoretically

So I recently rediscovered simulated annealing though a path that others seem to be well aware of. I was aware of Metropolis-Hastings as a sampling algorithm that creates a Markov-Chain who's ...
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1answer
52 views

How to calculate conditional entropy

I'm new to information theory and I am struggling to understand this problem. Let $p(x,y)$ given by: How can we calculate $H(X|Y)$? I know $H(X|Y)=H(X|Y=0)+H(X|Y=1)$ but then I don't know how to go ...
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1answer
100 views

Codeword constructed by Huffman's algorithm has average length of at most log n

I am interested in the following question: Prove that the average length of a codeword constructed by Huffman's algorithm has average length at most $\log n$, where $n$ is the number of codewords. ...
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1answer
67 views

Proof on lower bound of search in unsorted array with information theory?

I know there are proofs using an adversary technique. I've seen other proofs for search in a sorted list using information theory. But I haven't come across a proof using it to prove the lower bound ...
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32 views

Can we think of information theory in terms of “a measure on set of information”?

In information theory, we deal with the quantities $I(X;Y), H(X),H(Y), H(X|Y), H(Y|X)$. These are just numbers, but I intuitively think of them as the "measure" of a set of information. There is at ...
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1answer
51 views

How connected are information theory and algorithmic information theory?

In the book by Cover and Thomas on information theory, there is a chapter on algorithmic information theory (kolmogorov complexity and so forth). As far as I understand, algorithmic information ...
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2answers
67 views

What is a symbol code?

I am a physicist learning a bit of information theory. I have encountered a term ("symbol codes") on Wikipedia, and cannot find what it means: Source coding theorem for symbol codes Let $\...
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3answers
111 views

Is it possible to have high compression but low predictability?

Can you have a process that generates a binary sequence with high compression rate (low entropy) but impossible to predict next symbol? 'impossible to predict' - sequence cannot be predicted ...
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17 views

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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1answer
49 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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75 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
102 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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77 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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