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Dirichlet Average
Dirichlet averages are averages of functions under the Dirichlet distribution. An important one are dirichlet averages that have a certain argument structure, namely : F(\mathbf;\mathbf)=\int f( \mathbf \cdot \mathbf) \, d \mu_b(\mathbf), where \mathbf\cdot\mathbf=\sum_i^N u_i \cdot z_i and d \mu_b(\mathbf)=u_1^ \cdots u_N^ d\mathbf is the Dirichlet measure with dimension ''N''. They were introduced by the mathematician Bille C. Carlson in the '70s who noticed that the simple notion of this type of averaging generalizes and unifies many special functions, among them generalized hypergeometric functions or various orthogonal polynomials:. They also play an important role for the solution of elliptic integrals (see Carlson symmetric form) and are connected to statistical applications in various ways, for example in Bayesian analysis Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evid ... [...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] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \left(t, \sqrt \right) \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlson Symmetric Form
In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa. The Carlson elliptic integrals are: R_F(x,y,z) = \tfrac\int_0^\infty \frac R_J(x,y,z,p) = \tfrac\int_0^\infty \frac R_C(x,y) = R_F(x,y,y) = \tfrac \int_0^\infty \frac R_D(x,y,z) = R_J(x,y,z,z) = \tfrac \int_0^\infty \frac Since R_C and R_D are special cases of R_F and R_J, all elliptic integrals can ultimately be evaluated in terms of just R_F and R_J. The term ''symmetric'' refers to the fact that in contrast to the Legendre forms, these functions are unchanged by the exchange of certain subsets of their arguments. The value of R_F(x,y,z) is the same for any permutation of its arguments, and the value of R_J(x,y,z,p) is the same for any permutation of its first three arguments. The Carlson elliptic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Analysis
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Introduction to Bayes' rule Formal explanation Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem: P(H\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
