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Kirkwood Approximation
The Kirkwood superposition approximation was introduced in 1935 by John G. Kirkwood as a means of representing a discrete probability distribution. The Kirkwood approximation for a discrete probability density function P(x_,x_,\ldots ,x_) is given by : P^(x_1,x_2,\ldots ,x_n) = \prod_^\left prod_p(\mathcal_i)\right = \frac where : \prod_p(\mathcal_i) is the product of probabilities over all subsets of variables of size ''i'' in variable set \scriptstyle\mathcal. This kind of formula has been considered by Watanabe (1960) and, according to Watanabe, also by Robert Fano. For the three-variable case, it reduces to simply : P^\prime(x_1,x_2,x_3)=\frac The Kirkwood approximation does not generally produce a valid probability distribution (the normalization condition is violated). Watanabe claims that for this reason informational expressions of this type are not meaningful, and indeed there has been very little written about the properties of this measure. The Kirkwood approxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Gamble Kirkwood
John "Jack" Gamble Kirkwood (May 30, 1907, Gotebo, Oklahoma – August 9, 1959, New Haven, Connecticut) was a noted chemist and physicist, holding faculty positions at Cornell University, the University of Chicago, California Institute of Technology, and Yale University. Early life and background Kirkwood was born in Gotebo, Oklahoma, the oldest child of John Millard and Lillian Gamble Kirkwood. His father was educated as an attorney and was a distributor for the Goodyear Corporation in the state of Kansas. In addition to Jack Kirkwood, there were two younger sisters: Caroline (1910) and Margaret (1921). In 1909, the family moved to Wichita, Kansas. In the 1920s the family traveled to Pasadena, California to escape Midwestern winters. Education While in Pasadena, Kirkwood, age 15, audited chemistry classes at Caltech. Showing remarkable talent in mathematics and chemistry, Kirkwood was persuaded by A. A. Noyes to enroll at Caltech before finishing his high school education ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...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] |
Judea Pearl
Judea Pearl (born September 4, 1936) is an Israeli-American computer scientist and philosopher, best known for championing the probabilistic approach to artificial intelligence and the development of Bayesian networks (see the article on belief propagation). He is also credited for developing a theory of causal and counterfactual inference based on structural models (see article on causality). In 2011, the Association for Computing Machinery (ACM) awarded Pearl with the Turing Award, the highest distinction in computer science, "for fundamental contributions to artificial intelligence through the development of a calculus for probabilistic and causal reasoning". He is the author of several books, including the technical Causality: Models, Reasoning and Inference, and The Book of Why, a book on causality aimed at the general public. Judea Pearl is the father of journalist Daniel Pearl, who was kidnapped and murdered by terrorists in Pakistan connected with Al-Qaeda and the Inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
In the mathematical area of graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a Disjoint union of graphs, disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Distributions
Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a group with the discrete topology *Discrete category, category whose only arrows are identity arrows *Discrete mathematics, the study of structures without continuity *Discrete optimization, a branch of optimization in applied mathematics and computer science *Discrete probability distribution, a random variable that can be counted *Discrete space, a simple example of a topological space *Discrete spline interpolation, the discrete analog of ordinary spline interpolation *Discrete time, non-continuous time, which results in discrete-time samples *Discrete variable In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by ''measuring'' or ''counting'', respectively. If it can ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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