Questions tagged [information-theory]

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

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Unable to think about a question in coding theory

I am trying assignments of coding theory and I am unable to think about this problem. Question is -> Show that there doesn't exist a [12, 7,5] code. I have no idea how to obtain a contradiction. ...
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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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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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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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21 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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25 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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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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Information bottleneck function

On the Wikipedia page, the information bottleneck function is defined in the following way. For some random variable $X$ containing information about the relevant variable $Y$, we have some joint ...
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29 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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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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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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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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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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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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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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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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42 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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78 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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
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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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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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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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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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762 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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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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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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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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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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