|
Data Processing Inequality
The data processing inequality is an information theoretic concept that 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'. Statement 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 . That is, Z and Y contain the same information about X, and X \rightarrow Z \rightarrow Y also forms a Markov chain. Proof On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, Neuroscience, neurobiology, physics, 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, fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a dice, die (which has six equally likely outcomes). Some other important measu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
In probability theory and statistics, a Markov chain or Markov process is a stochastic process 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). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, fin ... [...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 estab ... [...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 Statistical dependence, dependence between the two variables. More specifically, it quantifies the "Information content, amount of information" (in Units of information, units such as shannon (unit), shannons (bits), Nat (unit), nats or Hartley (unit), hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of Entropy (information theory), 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 Pearson correlation coefficient, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Mutual Information
Conditional (if then) may refer to: *Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B *Conditional proof, in logic: a proof that asserts a conditional, and proves that the antecedent leads to the consequent *Material conditional, in propositional calculus, or logical calculus in mathematics *Relevance conditional, in relevance logic *Conditional (computer programming), a statement or expression in computer programming languages *A conditional expression in computer programming languages such as ?: *Conditions in a contract Grammar and linguistics *Conditional mood (or conditional tense), a verb form in many languages *Conditional sentence, a sentence type used to refer to hypothetical situations and their consequences **Indicative conditional, a conditional sentence expressing "if A then B" in a natural language **Counterfactual conditional, a conditional sentence indicating what would b ... [...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, biased or poor quality ("garbage") information or input (computer science), input produces a result or input/output, output of similar ("garbage") quality. The adage points to the need to improve data quality in, for example, programming. Rubbish in, rubbish out (RIRO) is an alternate wording. The principle applies to all logical Argumentation theory, argumentation: soundness implies validity (logic), 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 comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
