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Questions and Answers
What are the IT framework components listed in W.Henkel's content?
data source, channel, modulator, encoder, noise, demodulator, decoder
What are the main topics covered in W.Henkel's content?
In the basics of stochastics, the probability of an event x is approximately nx / n, where x is obtained nx times. What is x?
x ∈ X
Bayes' theorem states that the joint probability of two events equals the product of the conditional probabilities of each event.
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What is the definition of the entropy in information theory?
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What does the entropy achieve its maximum for, when the number of possible events (n) is fixed?
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What is the formula for entropy of repeated independent trials, also known as a memoryless source?
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Define the joint entropy H(X, Y) and the conditional entropy H(Y|X).
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The chain rule states that H(X, Y) is equal to H(X) multiplied by H(Y|X).
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What is a Markov source?
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What is the formula to calculate the steady-state probabilities in a Markov chain?
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What is the average number of steps required to return to state i equal to?
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In a Markov chain, states are said to communicate if P(Sn = j|S0 = i) > 0. Each Markov chain has a unique decomposition of the state space S into a sequence of disjoint subsets C1, C2, ..., S = ___.
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For determining a stationary state in a Markov chain, which of the following conditions need to be satisfied? (Select all that apply)
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In a Markov chain, positive recurrent states can be periodic or non-periodic.
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What is the average codeword length defined as?
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Which property ensures that no codeword is a prefix of another?
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The efficiency of a source code is the ratio of the entropy over the average length.
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According to Kraft's inequality, a prefix-free code with codeword lengths l1, l2,..., lk exists if and only if Σ(r^(-li)) <= __
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What is the type of source considered in source encoding?
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According to Shannon's Source Coding Theorem, for the probability of a perfect reconstruction approaching one, it is necessary and sufficient to use ____ bit/source symbol for encoding.
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What is the equipartition property formulated as in the content?
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In source encoding, all block codes are uniquely decodable.
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What does the Lempel-Ziv-Welch 84 algorithm do?
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Define Markov chain according to the provided content.
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In the formula provided, $Pen \geq 1 - C/R$, for $n \to \infty$ this will result in ___________.
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What is the differential entropy of a continuous random variable X with a density fX(x)?
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Define the mutual information between random variables X and Y.
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What is the formula for calculating H(X|Y)?
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According to the provided content, what does the differential entropy of a Gaussian-distributed random variable depend on?
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What is the lower limit for the error probability if H(X|Y) > 0?
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For ν = 2 and H(X|Y) = 1/2, what is the range for the probability Pe?
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In the context of the toss of a dice and the random variables X, Y, and Z, calculate H(X).
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Given (X1, X2, X3) as a vector-valued random variable with specific values and probabilities, determine H(X1, X2).
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Explain why a weather forecast that is only right 50% of the time, but still makes sense when assuming it's always sunny, is justifiable from an information-theoretic perspective.
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What is the formula shown for mutual information in the given content?
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Complete the mathematical expression: H(Y |X) = - ∫∫ fX (x)fY|X (y|x) log fY|X (y|x) dx dy.
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Study Notes
Introduction to Information Theory
- The course covers the basics of information theory, including data sources, channels, modulators, and demodulators.
- The topics include:
- Introduction to sources and channels, probabilities, and densities, information, and entropy.
- Entropy and mutual information, including the chain rule, Jensen's inequality, and Fano's inequality.
- Source coding principles, including Kraft's inequality, optimum codes,Huffman coding, and Shannon-Fano codes.
- Practical lossless source coding algorithms, including Lempel-Ziv and arithmetic coding.
- The discrete channel, including channel capacity and channel coding theorem.
- The Gaussian and bandlimited channel, including capacities for modulation alphabets.
- Random coding bound, error exponent, and cutoff rate.
- Rate-distortion theory, multiple access, and MIMO.
- Channel coding, including linear block codes, Reed-Solomon codes, convolutional codes, and turbo codes.
- Cryptology, including public-key encryption.
Basics of Stochastics
- A random variable is a variable that can take on different values according to some underlying probability distribution.
- The probability of an event is a non-negative real number that represents the likelihood of the event occurring.
- The probability of an event is denoted by P(Q) or p(X) and satisfies the following properties:
- ∀Q ∈ Q(X) : p(Q) ≥ 0
- P(X) = 1
- ∀Q, R ∈ Q(X) : Q ∩ R = {} : p(Q ∪ R) = p(Q) + p(R)
- The expectation of a function g(x) is denoted by E{g(X)} and is defined as the sum of the product of the values of the function and the corresponding probabilities.
- The expectation of X is denoted by E{X} and is a measure of the central tendency of the distribution.
- The second moment or quadratic mean of X is denoted by E{(X)2 } and is a measure of the spread of the distribution.
- The kth moment of X is denoted by E{(X)k } and is a measure of the distribution's shape.
Information and Entropy
- The information content of an event is a measure of how much information is gained when the event occurs.
- The information content is denoted by I(x) and is defined as the negative logarithm of the probability of the event.
- The entropy of a random variable is a measure of the average information content of the variable.
- The entropy is denoted by H(X) and is defined as the expectation of the information content of the variable.
- The entropy of a binary source is denoted by H(p) and is a function of the probability of the source.
- The entropy of a binary source has the following properties:
- H(p) = 0 if and only if p = 0 or p = 1
- H(p) is a concave function of p
- H(p) is symmetric around p = 1/2
- The entropy of a discrete memoryless source is defined as the entropy of the individual symbols.
Joint and Conditional Entropies
- The joint entropy of two random variables is denoted by H(X, Y) and is a measure of the total uncertainty of the variables.
- The conditional entropy of one random variable given another is denoted by H(Y|X) and is a measure of the uncertainty of Y given X.
- The chain rule for entropy states that H(X, Y) = H(X) + H(Y|X).
- The chain rule for a collection of random variables states that H(X1, X2, ..., Xn) = H(X1) + H(X2|X1) + ... + H(Xn|Xn-1, ..., X1).### Independence Bound on Entropy
- The independence bound on entropy is given by: H(X1, X2, ..., Xn) ≤ ∑ H(Xi)
- Equality holds if and only if the Xi are statistically independent.
- This is proved using the chain rule for entropies: H(X1, X2, ..., Xn) = ∑ H(Xi |Xi-1, ..., X1) ≤ ∑ H(Xi)
Markov Sources
- A Markov source is a source with memory m, where the probability of the next state depends on the previous m states.
- The probability of the next state is given by: P(x(i) | s(i)) = P(x(i) | x(i-1), x(i-2), ..., x(i-m))
- The state transitions are given by: s(i) = (x(i-1), x(i-2), ..., x(i-m)) → s(i+1) = (x(i), x(i-1), ..., x(i-m+1))
- The number of states is q^m, where q is the number of discrete elements.
Graphical Representation of Markov Sources
- Markov sources can be represented graphically using a state diagram.
- The transition probabilities are represented by the edges between states.
Stationary State Probabilities
- The stationary state probabilities are given by: ps = ps Î , where Î is the Markov matrix.
- The stationary state probabilities can be found by solving the equation: ps (U + Î - I) = 1, where U is a matrix of ones.
Special Cases of Markov Chains
- A Markov chain is said to be irreducible if it has only one communication class.
- A Markov chain is said to be periodic with period k if the probability of returning to a state is k.
- An absorbing state is a state with Pii = 1 and Pij = 0 for i ≠j.
Entropy Calculations for Markov Chains
- The average information content for a state sj is given by: H(X|sj) = -∑ P(xk|sj) log P(xk|sj)
- The average information content of the Markov chain over all states sj is given by: H(X) = ∑ P(sj) H(X|sj)
- The entropy of a Markov chain can be calculated using the formula: H(X) = -∑ P(xk, sj) log P(xk|sj)
Fano's Lemma
- Fano's lemma states that: H(X|Y) ≤ h(Pe) + Pe log2 (ν - 1), where Pe is the probability of error.
- The lemma gives an upper bound on the conditional entropy H(X|Y) in terms of the probability of error Pe.
Applications of Markov Chains
- Markov chains can be used to model English text, where the probability of a word depends on the previous words.
- Markov chains can be used to model a queuing problem, where the probability of a customer arriving or departing depends on the previous state of the queue.### Entropy and Information Theory
- Conditional entropy H(Y|X) is the uncertainty of the weather after receiving a forecast.
- H(Y|X) is calculated using Bayes' rule: P(y|x) = P(y,x)/P(x).
- In this example, H(Y|X) = 0.689, which is less than H(Y) = 0.81, indicating that the weather forecast reduces the uncertainty of the weather.
Jensen's Inequality
- Definition: A function f(x) is convex over an interval (a, b) if for every x1, x2 ∈ (a, b) and 0 ≤ λ ≤ 1, f(λx1 + (1 - λ)x2) ≤ λf(x1) + (1 - λ)f(x2).
- If a function is convex, it always lies below a straight line between the points (x1, f(x1)) and (x2, f(x2)).
- Examples of convex functions: x^2, |x|, e^x, x log x (for x ≥ 0).
- Theorem: If a function has a non-negative second derivative, it is convex.
Fisher Information, Rényi Entropy, and Gibbs Entropy
- Fisher information: J(θ) is the variance of the score, ∂/∂θ ln f(x; θ), where f(x; θ) is a probability density function.
- Rényi entropy: Hm(P1, ..., Pm) = 1/(1 - α) log (∑ Pi^α) for α > 0, α ≠1.
- Gibbs entropy (thermodynamics): S = -kB ∑ Pi ln Pi, where Pi is the probability of a microstate and kB is the Boltzmann constant.
Typical Sequences
- A typical sequence set has the highest probability, and its element sequences need not necessarily have higher probability than sequences in the non-typical set.
- The set Aε of ε-typical sequences x = x1, x2, ..., xn is defined as Aε(X) = {x : -log PX(x) - H(X) ≤ ε}.
The Asymptotic Equipartition Property (AEP)
- The AEP splits the set of sequences into typical and non-typical ones.
- Although the typical sequences are much fewer, their probability sums up to almost one.
- As the sequence length increases, the probability of an untypical sequence decreases.
Tschebyscheff's Inequality
- Proof: P(|x - µ| ≥ ε) ≤ σ^2 / ε^2, where σ^2 is the variance of the random variable X.
- Application: In the coin flipping experiment, σ^2 = pq / n, where p is the probability of heads and q is the probability of tails.
The Equipartition Property
- Theorem: For each ε > 0, there exists an n ∈ N, such that Aε(X) has the following properties:
- P(Aε(X)) ≥ 1 - ε
- x ∈ Aε(X) ⇒ -log PX(x) - H(X) ≤ ε
- (1 - ε)^2n(H(X) - ε) ≤ |Aε(X)| ≤ 2n(H(X) + ε)
Shannon's Source Coding Theorem
- For the probability of a perfect reconstruction approaching one, it is necessary and sufficient to use H(X) bits/source symbol for encoding.
- Proof: For an arbitrary ε > 0, there exists a sufficiently large n, such that a block of n source symbols X1, X2, ..., Xn can be uniquely encoded with n(H(X) + ε) bits, despite non-typical source sequences.
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Description
This quiz covers the basics of Information Theory, including the IT framework and its components such as data source, channel, and modulator. It also touches on encoding and decoding concepts.
