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Data Processing Inequality
The data processing inequality is an information theoretic concept which states that the information content of a signal cannot be increased via a local physical operation. This can be expressed concisely as 'post-processing cannot increase information'. Definition Let three random variables form the Markov chain X \rightarrow Y \rightarrow Z, implying that the conditional distribution of Z depends only on Y and is conditionally independent of X. Specifically, we have such a Markov chain if the joint probability mass function can be written as :p(x,y,z) = p(x)p(y, x)p(z, y)=p(y)p(x, y)p(z, y) In this setting, no processing of Y, deterministic or random, can increase the information that Y contains about X. Using the mutual information, this can be written as : : I(X;Y) \geqslant I(X;Z) With the equality I(X;Y) = I(X;Z) if and only if I(X;Y\mid Z)=0 , i.e. Z and Y contain the same information about X, and X \rightarrow Z \rightarrow Y also forms a Markov chain. Proof One can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering (field), information engineering, and electrical engineering. A key measure in information theory is information entropy, entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a dice, die (with six equally likely outcomes). Some other important measures in information theory are mutual informat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Independence
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probability, as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability without. If A is the hypothesis, and B and C are observations, conditional independence can be stated as an equality: :P(A\mid B,C) = P(A \mid C) where P(A \mid B, C) is the probability of A given both B and C. Since the probability of A given C is the same as the probability of A given both B and C, this equality expresses that B contributes nothing to the certainty of A. In this case, A and B are said to be conditionally independent given C, written symbolically as: (A \perp\!\!\!\perp B \mid C). The concept of conditional independence is essential to graph-based theories of statistical inference, as it establ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutual Information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such as shannons (bits), nats or hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable. Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair (X,Y) is from the product of the marginal distributions of X and Y. MI is the expected value of the pointwise mutual information (PMI). The quantity was defined and analyzed by Claude Shannon in his landmark paper "A Mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Mutual Information
In probability theory, particularly information theory, the conditional mutual information is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third. Definition For random variables X, Y, and Z with support sets \mathcal, \mathcal and \mathcal, we define the conditional mutual information as This may be written in terms of the expectation operator: I(X;Y, Z) = \mathbb_Z P_ \otimes P_ )/math>. Thus I(X;Y, Z) is the expected (with respect to Z) Kullback–Leibler divergence from the conditional joint distribution P_ to the product of the conditional marginals P_ and P_. Compare with the definition of mutual information. In terms of PMFs for discrete distributions For discrete random variables X, Y, and Z with support sets \mathcal, \mathcal and \mathcal, the conditional mutual information I(X;Y, Z) is as follows : I(X;Y, Z) = \sum_ p_Z(z) \sum_ \sum_ p_(x,y, z) \log \frac where the marginal, joint, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Mutual Information
In probability theory, particularly information theory, the conditional mutual information is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third. Definition For random variables X, Y, and Z with support sets \mathcal, \mathcal and \mathcal, we define the conditional mutual information as This may be written in terms of the expectation operator: I(X;Y, Z) = \mathbb_Z P_ \otimes P_ )/math>. Thus I(X;Y, Z) is the expected (with respect to Z) Kullback–Leibler divergence from the conditional joint distribution P_ to the product of the conditional marginals P_ and P_. Compare with the definition of mutual information. In terms of PMFs for discrete distributions For discrete random variables X, Y, and Z with support sets \mathcal, \mathcal and \mathcal, the conditional mutual information I(X;Y, Z) is as follows : I(X;Y, Z) = \sum_ p_Z(z) \sum_ \sum_ p_(x,y, z) \log \frac where the marginal, joint, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Garbage In, Garbage Out
In computer science, garbage in, garbage out (GIGO) is the concept that flawed, or nonsense (garbage) input data produces nonsense output. Rubbish in, rubbish out (RIRO) is an alternate wording. The principle applies to all logical argumentation: soundness implies validity, but validity does not imply soundness. History The expression was popular in the early days of computing. The first known use is in a 1957 syndicated newspaper article about US Army mathematicians and their work with early computers, in which an Army Specialist named William D. Mellin explained that computers cannot think for themselves, and that "sloppily programmed" inputs inevitably lead to incorrect outputs. The underlying principle was noted by the inventor of the first programmable computing device design: More recently, the Marine Accident Investigation Branch comes to a similar conclusion: The term may have been derived from last-in, first-out (LIFO) or first-in, first-out (FIFO). Uses This p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
