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Generalized Dirichlet Distribution
In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely neutral . The density function of p_1,\ldots,p_ is : \left \prod_^B(a_i,b_i)\right p_k^ \prod_^\left p_i^\left(\sum_^kp_j\right)^\right where we define p_k= 1- \sum_^p_i. Here B(x,y) denotes the Beta function. This reduces to the standard Dirichlet distribution if b_=a_i+b_i for 2\leqslant i\leqslant k-1 (b_0 is arbitrary). For example, if ''k=4'', then the density function of p_1,p_2,p_3 is : \left prod_^B(a_i,b_i)\right p_1^p_2^p_3^p_4^\left(p_2+p_3+p_4\right)^\left(p_3+p_4\right)^ where p_1+p_2+p_3<1 and . Connor and Mosimann define the PDF as they did for the following reason. Define random variables with |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \boldsymbol\alpha of positive reals. It is a multivariate generalization of the beta distribution, (Chapter 49: Dirichlet and Inverted Dirichlet Distributions) hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution. The infinite-dimensional generalization of the Dirichlet distribution is the ''Dirichlet process''. Definitions Probability density function The Dirichlet distribution of order ''K'' ≥ 2 with parameters ''α''1, ..., ''α''''K'' > 0 has a probability density function with respect to Lebesgue m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neutral Vector
In statistics, and specifically in the study of the Dirichlet distribution, a neutral vector of random variables is one that exhibits a particular type of statistical independence amongst its elements. In particular, when elements of the random vector must add up to certain sum, then an element in the vector is neutral with respect to the others if the distribution of the vector created by expressing the remaining elements as proportions of their total is independent of the element that was omitted. Definition A single element X_i of a random vector X_1,X_2,\ldots,X_k is neutral if the ''relative'' proportions of all the other elements are independent of X_i. Formally, consider the vector of random variables :X=(X_1,\ldots,X_k) where :\sum_^k X_i=1. The values X_i are interpreted as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say X_1, and consider the distribution of the remaining intervals within the remaining length. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \Re(z_1), \Re(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.Davis (1972) 6.2.2 p.258 A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac. A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function thatDavis (1972) 6.2.1 p.258 : \Beta(m,n) =\dfrac = \frac \B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the random variable and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas ''beta distribution of the second kind'' is an alternative name for the beta prime distribution. The generalization to mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multinomial Distribution
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a ''k''-sided dice rolled ''n'' times. For ''n'' independent trials each of which leads to a success for exactly one of ''k'' categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories. When ''k'' is 2 and ''n'' is 1, the multinomial distribution is the Bernoulli distribution. When ''k'' is 2 and ''n'' is bigger than 1, it is the binomial distribution. When ''k'' is bigger than 2 and ''n'' is 1, it is the categorical distribution. The term "multinoulli" is sometimes used for the categorical distribution to emphasize this four-way relationship (so ''n'' determines the prefix, and ''k'' the suffix). The Bernoulli distribution models the outcome of a single Bernoulli trial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet-multinomial Distribution
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution (after George Pólya). It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector \boldsymbol, and an observation drawn from a multinomial distribution with probability vector p and number of trials ''n''. The Dirichlet parameter vector captures the prior belief about the situation and can be seen as a pseudocount: observations of each outcome that occur before the actual data is collected. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, machine learning, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution. It reduces to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lukacs's Proportion-sum Independence Theorem
In statistics, Lukacs's proportion-sum independence theorem is a result that is used when studying proportions, in particular the Dirichlet distribution. It is named after Eugene Lukacs. The theorem If ''Y''1 and ''Y''2 are non-degenerate, independent random variables, then the random variables : W=Y_1+Y_2\textP = \frac are independently distributed if and only if both ''Y''1 and ''Y''2 have gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...s with the same scale parameter. Corollary Suppose ''Y'' ''i'', ''i'' = 1, ..., ''k'' be non-degenerate, independent, positive random variables. Then each of ''k'' − 1 random variables : P_i=\frac is independent of : W=\sum_^k Y_i if and only if all the ''Y'' ''i'' hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Distributions
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of Continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the conic secti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Continuous Distributions
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Multivariate may refer to: In mathematics * Multivariable calculus * Multivariate function * Multivariate polynomial In computing * Multivariate cryptography * Multivariate division algorithm * Multivariate interpolation * Multivariate optical computing * Multivariate optimization, used for the design of heat exchangers, see In statistics * Multivariate analysis * Multivariate random variable * Multivariate statistics See also * Univariate * Bivariate (other) Bivariate may refer to: Mathematics * Bivariate function, a function of two variables * Bivariate polynomial, a polynomial of two indeterminates Statistics * Bivariate data, that shows the relationship between two variables * Bivariate analys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Prior Distributions
Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change of sign of the imaginary part of a complex number * Conjugate (square roots), the change of sign of a square root in an expression *Conjugate element (field theory), a generalization of the preceding conjugations to roots of a polynomial of any degree *Conjugate transpose, the complex conjugate of the transpose of a matrix *Harmonic conjugate in complex analysis * Conjugate (graph theory), an alternative term for a line graph, i.e. a graph representing the edge adjacencies of another graph *In group theory, various notions are called conjugation: **Inner automorphism, a type of conjugation homomorphism **Conjugation in group theory, related to matrix similarity in linear algebra **Conjugation (group theory), the image of an element under t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
