|
Dual Total Correlation
In information theory, dual total correlation (Han 1978), information rate (Dubnov 2006), excess entropy (Olbrich 2008), or binding information (Abdallah and Plumbley 2010) is one of several known non-negative generalizations of mutual information. While total correlation is bounded by the sum entropies of the ''n'' elements, the dual total correlation is bounded by the joint-entropy of the ''n'' elements. Although well behaved, dual total correlation has received much less attention than the total correlation. A measure known as "TSE-complexity" defines a continuum between the total correlation and dual total correlation (Ay 2001). Definition For a set of ''n'' random variables \, the dual total correlation D(X_1,\ldots,X_n) is given by : D(X_1,\ldots,X_n) = H\left( X_1, \ldots, X_n \right) - \sum_^n H\left( X_i \mid X_1, \ldots, X_, X_, \ldots, X_n \right) , where H(X_,\ldots,X_) is the joint entropy of the variable set \ and H(X_i \mid \cdots ) is the conditional entropy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...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] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Entropy
In information theory, joint entropy is a measure of the uncertainty associated with a set of variables. Definition The joint Shannon entropy (in bits) of two discrete random variables X and Y with images \mathcal X and \mathcal Y is defined as where x and y are particular values of X and Y, respectively, P(x,y) is the joint probability of these values occurring together, and P(x,y) \log_2 (x,y)/math> is defined to be 0 if P(x,y)=0. For more than two random variables X_1, ..., X_n this expands to where x_1,...,x_n are particular values of X_1,...,X_n, respectively, P(x_1, ..., x_n) is the probability of these values occurring together, and P(x_1, ..., x_n) \log_2 (x_1, ..., x_n)/math> is defined to be 0 if P(x_1, ..., x_n)=0. Properties Nonnegativity The joint entropy of a set of random variables is a nonnegative number. :\Eta(X,Y) \geq 0 :\Eta(X_1,\ldots, X_n) \geq 0 Greater than individual entropies The joint entropy of a set of variables is greater than or eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Entropy
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable Y given that the value of another random variable X is known. Here, information is measured in shannons, nats, or hartley Hartley may refer to: Places Australia *Hartley, New South Wales *Hartley, South Australia **Electoral district of Hartley, a state electoral district Canada *Hartley Bay, British Columbia United Kingdom *Hartley, Cumbria *Hartley, Plymou ...s. The ''entropy of Y conditioned on X'' is written as \Eta(Y, X). Definition The conditional entropy of Y given X is defined as where \mathcal X and \mathcal Y denote the support sets of X and Y. ''Note:'' Here, the convention is that the expression 0 \log 0 should be treated as being equal to zero. This is because \lim_ \theta\, \log \theta = 0. Intuitively, notice that by definition of Expected value, expected value and of Conditional Probability, conditional proba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interaction Information
The interaction information is a generalization of the mutual information for more than two variables. There are many names for interaction information, including ''amount of information'', ''information correlation'', ''co-information'', and simply ''mutual information''. Interaction information expresses the amount of information (redundancy or synergy) bound up in a set of variables, ''beyond'' that which is present in any subset of those variables. Unlike the mutual information, the interaction information can be either positive or negative. These functions, their negativity and minima have a direct interpretation in algebraic topology. Definition The conditional mutual information can be used to inductively define the interaction information for any finite number of variables as follows: :I(X_1;\ldots;X_) = I(X_1;\ldots;X_n) - I(X_1;\ldots;X_n\mid X_), where :I(X_1;\ldots;X_n \mid X_) = \mathbb E_\big(I(X_1;\ldots;X_n) \mid X_\big). Some authors define the interaction inf ... [...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] |
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] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Probability Theory")
