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Probability distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted Dir ( α ) {\displaystyle \operatorname
Dirichlet_distribution
Distributions in probability theory
theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative
Dirichlet-multinomial distribution
Dirichlet-multinomial_distribution
Family of stochastic processes
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes
Dirichlet_process
Probability distribution
In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure
Generalized Dirichlet distribution
Generalized_Dirichlet_distribution
Definition and first properties of the Poisson-Dirichlet distributions
In probability theory, Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-increasing sequences with sum 1, depending
Poisson-Dirichlet distribution
Poisson-Dirichlet_distribution
Discrete probability distribution
the Dirichlet-multinomial distribution as the binomial and beta distributions are univariate versions of the multinomial and Dirichlet distributions respectively
Beta-binomial_distribution
Probability distribution
multiple variables is called a Dirichlet distribution. The probability density function (PDF) of the beta distribution, for 0 ≤ x ≤ 1 {\displaystyle 0\leq
Beta_distribution
inverse Dirichlet distribution is a derivation of the matrix variate Dirichlet distribution. It is related to the inverse Wishart distribution. Suppose
Inverse Dirichlet distribution
Inverse_Dirichlet_distribution
inverted Dirichlet distribution is a multivariate generalization of the beta prime distribution, and is related to the Dirichlet distribution. It was first
Inverted Dirichlet distribution
Inverted_Dirichlet_distribution
Stochastic process in probability theory
drawn from G0, with weights drawn from a two-parameter Poisson-Dirichlet distribution. The process is named after Jim Pitman and Marc Yor. The parameters
Pitman–Yor_process
Discrete probability distribution
It connects the categorical distribution with the related multinomial distribution. It shows why the Dirichlet distribution is the conjugate prior of the
Categorical_distribution
Numerical parameter in probability theory
conjunction with distributions whose domain is a probability distribution, such as the symmetric Dirichlet distribution and the Dirichlet process. The rest
Concentration_parameter
Dirichlet distribution (probability theory) Grouped Dirichlet distribution Inverted Dirichlet distribution Matrix variate Dirichlet distribution Dirichlet divisor
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Probability distribution
{1}{x_{i}(1-x_{i})}}} . The logistic normal distribution is a more flexible alternative to the Dirichlet distribution in that it can capture correlations between
Logit-normal_distribution
Probability multivariate distribution
probability theory and statistics, the Dirichlet negative multinomial distribution is a multivariate distribution on the non-negative integers. It is a
Dirichlet negative multinomial distribution
Dirichlet_negative_multinomial_distribution
German mathematician (1805–1859)
Johann Peter Gustav Lejeune Dirichlet (/ˌdɪərɪˈkleɪ/; German: [ləˈʒœn diʁiˈkleː]; 13 February 1805 – 5 May 1859) was a German mathematician. In number
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Probability distribution
In statistics, the grouped Dirichlet distribution (GDD) is a multivariate generalization of the Dirichlet distribution It was first described by Ng et
Grouped Dirichlet distribution
Grouped_Dirichlet_distribution
Generalization of the binomial distribution
_{i=1}^{k}p_{i}^{x_{i}}.} This form shows its resemblance to the Dirichlet distribution, which is its conjugate prior. Suppose that in a three-way election
Multinomial_distribution
Generative topic model
In natural language processing, latent Dirichlet allocation (LDA) is a generative statistical model that explains how a collection of text documents can
Latent_Dirichlet_allocation
Mathematical function for the probability a given outcome occurs in an experiment
normal distribution, etc. Dirichlet distribution, for a vector of probabilities that must sum to 1; conjugate to the categorical distribution and multinomial
Probability_distribution
Probability distribution
Xn, follows a Dirichlet distribution with parameters α1, ..., αn. For large α the gamma distribution converges to normal distribution with mean μ = αθ
Gamma_distribution
joint distribution is the product of their individual density functions. The Dirichlet distribution, a generalization of the beta distribution. The Ewens's
List of probability distributions
List_of_probability_distributions
Topics referred to by the same term
variable that exhibits a particular type of statistical independence (Dirichlet distribution) Neutrality (philosophy), the absence of declared or intentional
Neutral
Integral of sin(x)/x from 0 to infinity
are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral
Dirichlet_integral
Discrete-time stochastic process
from the original on 2012-09-25. Retrieved 2011-05-11. "Dirichlet Process and Dirichlet Distribution -- Polya Restaurant Scheme and Chinese Restaurant Process"
Chinese_restaurant_process
the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution and of the Dirichlet distribution. Suppose U 1 , …
Matrix variate Dirichlet distribution
Matrix_variate_Dirichlet_distribution
Concept in probability theory
The same issues apply to the Dirichlet distribution. β is rate or inverse scale. In parameterization of gamma distribution,θ = 1/β and k = α. This is the
Conjugate_prior
Probability distribution
no closed form version for the Beta-prime distribution except in special cases. The Dirichlet Distribution of order K ≥ 2 {\displaystyle K\geq 2} with
Distribution of the product of two random variables
Distribution_of_the_product_of_two_random_variables
Problem of solving a partial differential equation subject to prescribed boundary values
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region
Dirichlet_problem
Statistical Markov model
prior Dirichlet distribution, in which one Dirichlet distribution (the upper distribution) governs the parameters of another Dirichlet distribution (the
Hidden_Markov_model
Statistical technique for smoothing categorical data
expected value of the posterior distribution, using a symmetric Dirichlet distribution with parameter α as a prior distribution. In the special case where
Additive_smoothing
Theorem in combinatorics
commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the
Pigeonhole_principle
Monte Carlo algorithm
Dirichlet prior, and the joint distribution of these variables after collapsing is a Dirichlet-multinomial distribution. The conditional distribution
Gibbs_sampling
Smooth approximation of one-hot arg max
discrete-discrete distribution needs to be mimicked in a differentiable manner. Softplus Multinomial logistic regression Dirichlet distribution – an alternative
Softmax_function
Statistical concept
K-dimensional random vector drawn from a Dirichlet distribution (the conjugate prior of the categorical distribution), and the parameters will be distributed
Mixture_model
Theorem on the number of primes in arithmetic sequences
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
In statistics, and specifically in the study of the Dirichlet distribution, a neutral vector of random variables is one that exhibits a particular type
Neutral_vector
Mathematical methods used in Bayesian inference and machine learning
{\displaystyle \alpha _{0}} . The Dirichlet distribution is the conjugate prior of the categorical distribution or multinomial distribution. W ( ) {\displaystyle
Variational_Bayesian_methods
Topics referred to by the same term
Gaussian distribution, also called the normal distribution, an important family of continuous probability distributions Generalized Dirichlet distribution, a
GD
Concept in statistics
beta distribution. Compounding a multinomial distribution with probability vector distributed according to a Dirichlet distribution yields a Dirichlet-multinomial
Compound probability distribution
Compound_probability_distribution
group of data, with the Dirichlet processes for all groups sharing a base distribution which is itself drawn from a Dirichlet process. This method allows
Hierarchical Dirichlet process
Hierarchical_Dirichlet_process
Theorem about independent random variables
result that is used when studying proportions, in particular the Dirichlet distribution. It is named after Eugene Lukacs. If Y1 and Y2 are non-degenerate
Lukacs's proportion-sum independence theorem
Lukacs's_proportion-sum_independence_theorem
Formula in probability theory
The Dirichlet distribution is the conjugate prior for the multinomial distribution, which means that the posterior distribution is also a Dirichlet distribution
Rule_of_succession
Probability distribution
{\tfrac {1}{\gamma }},1)} . The inverted Dirichlet distribution is a generalization of the beta prime distribution. If X ∼ β ′ ( α , β ) {\displaystyle X\sim
Beta_prime_distribution
Topics referred to by the same term
individuals in the dataset Dirichlet process, a stochastic process corresponding to an infinite generalization of the Dirichlet distribution Dynamic programming
DP
a Dirichlet prior requires only adding the outcome frequencies to the Dirichlet prior alpha values, resulting in a Dirichlet posterior distribution for
Expected value of sample information
Expected_value_of_sample_information
Generalization of the one-dimensional normal distribution to higher dimensions
statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional
Multivariate normal distribution
Multivariate_normal_distribution
Generalized function whose value is zero everywhere except at zero
the Dirichlet kernel restricted to the interval [−π,π] tends to a multiple of the delta function as N → ∞. This is interpreted in the distribution sense
Dirac_delta_function
Averages of functions under the Dirichlet distribution
Dirichlet averages are averages of functions under the Dirichlet distribution. An important one are dirichlet averages that have a certain argument structure
Dirichlet_average
Concept in mathematical analysis
In mathematical analysis, the Dirichlet kernel, is the collection of periodic functions defined as D n ( x ) = ∑ k = − n n e i k x = 1 + 2 ∑ k = 1 n cos
Dirichlet_kernel
Type of plane partition
Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons
Voronoi_diagram
Probability distribution
b=1,p,q)} , the multivariate inverted beta and inverted Dirichlet (Dirichlet type 2) distribution given by M G B 2 ( y ; a = 1 , b = 1 , p , q ) {\displaystyle
Generalized_beta_distribution
Concept in probability theory and statistics
hypergeometric, multivariate negative hypergeometric, multinomial, or Dirichlet distribution, but not in general otherwise. Formally, the partial correlation
Partial_correlation
Non-informative prior distribution
the Jeffreys prior for γ → {\textstyle {\vec {\gamma }}} is the Dirichlet distribution with all (alpha) parameters set to one half. This amounts to using
Jeffreys_prior
Probability distribution
Negative binomial distribution Multinomial distribution Inverted Dirichlet distribution, a conjugate prior for the negative multinomial Dirichlet negative multinomial
Negative multinomial distribution
Negative_multinomial_distribution
Generating pseudo-random numbers that follow a probability distribution
random variables Beta distribution#Random variate generation Dirichlet distribution#Random variate generation Exponential distribution#Random variate generation
Non-uniform random variate generation
Non-uniform_random_variate_generation
Mathematical identity used to evaluate certain improper integrals
mathematics, Dirichlet integrals play an important role in distribution theory. We can see the Dirichlet integral in terms of distributions. One of those
Lobachevsky_integral_formula
Poincaré–Steklov operator is called the Dirichlet to Neumann (DtN) operator. The values of the temperature on the surface is the Dirichlet boundary condition of the
Poincaré–Steklov_operator
Direct relationship Directional statistics Dirichlet distribution Dirichlet-multinomial distribution Dirichlet process Disattenuation Discrepancy function
List_of_statistics_articles
Statistics and machine learning technique
of possible ensembles (with model weights drawn randomly from a Dirichlet distribution having uniform parameters). This modification overcomes the tendency
Ensemble_learning
Analytic function in mathematics
Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. The Riemann zeta function
Riemann_zeta_function
Type of probability distribution
distributions". arXiv:2308.01749 [math.PR]. Marchal, Olivier; Arbel, Julyan (2017). "On the sub-Gaussianity of the Beta and Dirichlet distributions"
Sub-Gaussian_distribution
Probability distribution used in multivariate hypothesis testing
Chi-squared distribution Dirichlet distribution F-distribution Gamma distribution Hotelling's T-squared distribution Student's t-distribution Wishart distribution
Wilks's_lambda_distribution
Mathematical function
This multivariate beta function is used in the definition of the Dirichlet distribution. Its relationship to the beta function is analogous to the relationship
Beta_function
Generalization of beta distribution
The distribution of the largest eigenvalue is well approximated by a transform of the Tracy–Widom distribution. Matrix variate Dirichlet distribution (Potters
Matrix variate beta distribution
Matrix_variate_beta_distribution
Shape with three equal sides
four-dimensional Cartesian integer coordinates. Clifton Cathedral Dirichlet distribution Eternity puzzle Almost-equilateral Heronian triangle Hofstadter
Equilateral_triangle
Test for series convergence
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence
Dirichlet's_test
Formal power series
Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require
Generating_function
Topics referred to by the same term
queue Kingman's law - special case of the law governing a Poisson-Dirichlet distribution King's Men (disambiguation) Kingsman (disambiguation) Kingman's
Kingman
Family of probability distributions related to the normal distribution
gamma chi-squared beta Dirichlet Bernoulli categorical Poisson Wishart inverse Wishart geometric A number of common distributions are exponential families
Exponential_family
Multi-dimensional generalization of triangle
space is often used to represent the space of probability distributions. The Dirichlet distribution, for instance, is defined on a simplex. In industrial
Simplex
Type of probabilistic logic
represented as a Dirichlet PDF (Probability Density Function). Through the correspondence between opinions and Beta/Dirichlet distributions, subjective logic
Subjective_logic
Commonly used representation of patterns in biological sequences
This is equivalent to multiplying each column of the PPM by a Dirichlet distribution and allows the probability to be calculated for new sequences (that
Position_weight_matrix
Evaluates how likely it is that any difference between data sets arose by chance
routine check-ups. In Bayesian statistics, one would instead use a Dirichlet distribution as conjugate prior. If one took a uniform prior, then the maximum
Pearson's_chi-squared_test
Distribution of new data marginalized over the posterior
beta-binomial distribution and Dirichlet-multinomial distribution are all predictive distributions of exponential-family distributions (the normal distribution, binomial
Posterior predictive distribution
Posterior_predictive_distribution
Electoral system for two-tier voting bodies
related to voting theory as a consequence of an averaging over the Dirichlet distribution. Mathematics portal Political science portal List of countries and
Jagiellonian_compromise
Type of mathematical function
hyperbolic secant distribution, the Wishart distribution, if n ≥ p + 1, the Dirichlet distribution, if all parameters are ≥ 1, the gamma distribution if the shape
Logarithmically concave function
Logarithmically_concave_function
is named after Peter Gustav Lejeune Dirichlet. In many applications we want to model a collection of distributions such as the one used to represent temporal
Dependent_Dirichlet_process
Meromorphic function on the complex plane
accurate estimates of the distribution of prime numbers. The closest relatives of the Riemann zeta function are the Dirichlet L-functions, which include
L-function
Technique in statistics
Multidimensional Functions Based on Natural Gradient Descent with Dirichlet Distributions". Mathematics. 10 (19): 3556. doi:10.3390/math10193556. ISSN 2227-7390
Information_geometry
Summatory function of the divisor-counting function
Zbl 1065.11079. Heath-Brown, D. R. (1992). "The distribution and moments of the error term in the Dirichlet divisor problem". Acta Arithmetica. 60 (4): 389–415
Divisor_summatory_function
Stratification of a genetic population based on allele frequencies
Monte Carlo, modelling allele frequencies at each locus with a Dirichlet distribution. Since then, algorithms such as ADMIXTURE (developed by David Alexander
Population structure (genetics)
Population_structure_(genetics)
Cell-free DNA profiling method
parameter for Dirichlet distribution generation. The initial parameter for Dirichlet distribution is set to 20. From the obtained Dirichlet distribution, 2000
EPIC-Seq
Bayesian nonparametric model of probability distributions
was introduced by Thomas Ferguson as a prior over probability distributions. A Dirichlet process D P ( s , G 0 ) {\displaystyle \mathrm {DP} \left(s,G_{0}\right)}
Imprecise_Dirichlet_process
Exploring properties of the integers with complex analysis
the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem
Analytic_number_theory
French mathematician (1949–2014)
Heidelberg. Pitman, J., & Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. The Annals of Probability, 25(2)
Marc_Yor
empirical levels have been shown to follow a Dirichlet distribution, whose marginals are beta distribution. The scenario approach with L 1 {\displaystyle
Scenario_optimization
Second-order partial differential equation
change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem. The Neumann boundary
Laplace's_equation
Emeritus Professor of Statistics and Mathematics
Pitman, Jim; Yor, Marc (1997-04-01). "The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator". The Annals of Probability.
Jim_Pitman
Conjecture on zeros of the zeta function
this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation ( 1 − 2 2 s ) ζ ( s ) = η ( s ) = ∑
Riemann_hypothesis
Mathematical function
controlled by the Dickman function Golomb–Dickman constant Poisson-Dirichlet distribution Dickman, K. (1930). "On the frequency of numbers containing prime
Dickman_function
Compound probability distribution
\beta } red balls are observed. Negative binomial distribution Dirichlet negative multinomial distribution Johnson et al. (1993) Johnson, N.L.; Kotz, S.;
Beta negative binomial distribution
Beta_negative_binomial_distribution
Vector with non-negative entries that add up to one
super-exponentially small as n {\displaystyle n} increases. Stochastic matrix Dirichlet distribution Bertsekas, D. P., & Tsitsiklis, J. N. (2008). Introduction to Probability
Probability_vector
Inputs for which a function's value is non-zero
if f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } is the Dirichlet function that is 0 {\displaystyle 0} on irrational numbers and 1 {\displaystyle
Support_(mathematics)
Characterization of how many integers are prime
example is the distribution of the last digit of prime numbers. Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states
Prime_number_theorem
Type of statistical analysis
assumptions about the distribution of model residuals. nonparametric hierarchical Bayesian models, such as models based on the Dirichlet process, which allow
Nonparametric_statistics
Mixture of discrete and continuous distributions
sampling solution, where the factors follow a Dirichlet process mixture of rectified Gaussian distribution, and applied it in computational biology for
Rectified Gaussian distribution
Rectified_Gaussian_distribution
Family of stochastic optimization methods
Estimation of distribution algorithms (EDAs), sometimes called probabilistic model-building genetic algorithms (PMBGAs), are stochastic optimization methods
Estimation of distribution algorithm
Estimation_of_distribution_algorithm
Automated recognition of patterns and regularities in data
empirical observations – using e.g., the Beta- (conjugate prior) and Dirichlet-distributions. The Bayesian approach facilitates a seamless intermixing between
Pattern_recognition
Mathematics
is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed
Neumann_boundary_condition
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DIRICHLET DISTRIBUTION
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DIRICHLET DISTRIBUTION
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DIRICHLET DISTRIBUTION
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