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Generalization of the binomial theorem to other polynomials
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization
Multinomial_theorem
Topics referred to by the same term
Multinomial may refer to: Multinomial theorem, and the multinomial coefficient Multinomial distribution Multinomial logistic regression Multinomial test
Multinomial
Algebraic expansion of powers of a binomial
algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ( x
Binomial_theorem
Generalization of the binomial distribution
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts
Multinomial_distribution
later rediscovered by Euler, is a very simple application of the multinomial theorem, which states ( x 1 + x 2 + ⋯ + x m ) n = ∑ k 1 , k 2 , … , k m k
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
Milliken's tree theorem (Ramsey theory) Multinomial theorem (algebra, combinatorics) Mycielski's theorem (graph theory) Nicomachus's theorem (number theory)
List_of_theorems
representation of an integer Mahler's theorem Multinomial distribution Multinomial coefficient, Multinomial formula, Multinomial theorem Multiplicities of entries
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree
Taylor's_theorem
Theorem related to ordinary least squares
In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest
Gauss–Markov_theorem
Regression for more than two discrete outcomes
In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more
Multinomial logistic regression
Multinomial_logistic_regression
Distributions in probability theory
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite
Dirichlet-multinomial distribution
Dirichlet-multinomial_distribution
Machine learning kernel function
dimensions; for σ = 1 {\displaystyle \sigma =1} , its expansion using the multinomial theorem is: exp ( − 1 2 ‖ x − x ′ ‖ 2 ) = exp ( 2 2 x ⊤ x ′ − 1 2 ‖ x
Radial_basis_function_kernel
Statement in probability theory
probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker
Donsker's_theorem
Describes the highest power of primes dividing a binomial coefficient
{S_{2}(3)+S_{2}(7)-S_{2}(10)}{2-1}}={\dfrac {2+3-2}{2-1}}=3.} Kummer's theorem can be generalized to multinomial coefficients ( n m 1 , … , m k ) = n ! m 1 ! ⋯ m k ! {\displaystyle
Kummer's_theorem
Machine learning kernel function
the quadratic kernel. After using the multinomial theorem (twice—the outermost application is the binomial theorem) and regrouping, K ( x , y ) = ( ∑ i
Polynomial_kernel
If there are more items than boxes holding them, one box must contain at least two items
approximation theorem Hilbert's paradox of the Grand Hotel Multinomial theorem Pochhammer symbol Ramsey's theorem Herstein 1964, p. 90 Rittaud, Benoît; Heeffer, Albrecht
Pigeonhole_principle
Graphical aid for deriving some concepts in combinatorics
dots and dividers) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve a variety of counting problems, such as how many
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
French mathematician (1667–1754)
de Moivre also generalised Newton's noteworthy binomial theorem into the multinomial theorem. The Royal Society became apprised of this method in 1697
Abraham_de_Moivre
Probabilistic classification algorithm
With a multinomial event model, samples (feature vectors) represent the frequencies with which certain events have been generated by a multinomial ( p 1
Naive_Bayes_classifier
Theorem on the largest antichain of sets
Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family
Sperner's_theorem
Mathematical notation
(or R n → R {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} } ). Multinomial theorem ( ∑ i = 1 n x i ) k = ∑ | α | = k ( k α ) x α {\displaystyle \left(\sum
Multi-index_notation
Class of statistical models
(Y=m\mid Y\in \{1,m\}).\,} for m > 2. Different links g lead to multinomial logit or multinomial probit models. These are more general than the ordered response
Generalized_linear_model
Theorem about independent random variables
doi:10.1214/aoms/1177728549. Mosimann, James E. (1962). "On the compound multinomial distribution, the multivariate β {\displaystyle \beta } distribution
Lukacs's proportion-sum independence theorem
Lukacs's_proportion-sum_independence_theorem
Generalized chain rule in calculus
obtained by collecting like terms, or alternatively, by applying the multinomial theorem. The special case f ( x ) = e x {\displaystyle f(x)=e^{x}} , g (
Faà_di_Bruno's_formula
Mathematical set with repetitions allowed
coefficients should not be confused with the multinomial coefficients that occur in the multinomial theorem. The value of multiset coefficients can be given
Multiset
Concept in mathematics
{red}{6}}xy^{5}+{\color {red}1}y^{6}\,} Polynomial factorization Factorization Multinomial theorem Discussion Review of Algebra: Expansion Archived 2014-12-10 at the
Polynomial_expansion
Number of subsets of a given size
coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥
Binomial_coefficient
Probability distribution
distribution is the conjugate prior of the categorical distribution and multinomial distribution. The infinite-dimensional generalization of the Dirichlet
Dirichlet_distribution
In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that
Multinomial_probit
Regularization technique for ill-posed problems
the a priori distribution of x {\displaystyle x} , according to Bayes' theorem. If the assumption of normality is replaced by assumptions of homoscedasticity
Ridge_regression
Choice between two or more discrete alternatives
many forms, including: Binary Logit, Binary Probit, Multinomial Logit, Conditional Logit, Multinomial Probit, Nested Logit, Generalized Extreme Value Models
Discrete_choice
Evaluates how likely it is that any difference between data sets arose by chance
consequence of the Binomial theorem. The result about the numbers of degrees of freedom is valid when the original data are multinomial and hence the estimated
Pearson's_chi-squared_test
Type of probabilistic logic
and can be represented as a Beta PDF (Probability Density Function). A multinomial opinion applies to a state variable of multiple possible values, and
Subjective_logic
German mathematician
combinatorisch-analytischer Abhandlungen, which contained a claim that de Moivre's multinomial theorem was “the most important proposition in all of mathematical analysis”
Carl_Hindenburg
Branch of discrete mathematics
none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory. The types of questions
Combinatorics
Statistical model for a binary dependent variable
dog, lion, etc.), and the binary logistic regression generalized to multinomial logistic regression. If the multiple categories are ordered, one can
Logistic_regression
Statistical inequality
minimax character of the sample distribution function and of the classical multinomial estimator", Annals of Mathematical Statistics, 27 (3): 642–669, doi:10
Dvoretzky–Kiefer–Wolfowitz inequality
Dvoretzky–Kiefer–Wolfowitz_inequality
analysis Multinomial distribution Multinomial logistic regression Multinomial logit – see Multinomial logistic regression Multinomial probit Multinomial test
List_of_statistics_articles
Method for estimating the unknown parameters in a linear regression model
residuals when regressors have finite fourth moments and—by the Gauss–Markov theorem—optimal in the class of linear unbiased estimators when the errors are
Ordinary_least_squares
Dirichlet distribution (probability theory) Dirichlet-multinomial distribution Dirichlet negative multinomial distribution Generalized Dirichlet distribution
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Probability distribution and special case of gamma distribution
binomial, and instead require 3 or more categories, which leads to the multinomial distribution. Just as de Moivre and Laplace sought for and found the
Chi-squared_distribution
Probability distribution of energy states of a system
economic contexts. The Boltzmann distribution has the same form as the multinomial logit model. As a discrete choice model, this is very well known in economics
Boltzmann_distribution
Family of probability distributions related to the normal distribution
fixed and known. For example: binomial (with fixed number of trials) multinomial (with fixed number of trials) negative binomial (with fixed number of
Exponential_family
Generalization of the product rule in calculus
k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}} are the multinomial coefficients. This is akin to the multinomial formula from algebra. The proof of the general
General_Leibniz_rule
Discrete probability distribution
{\displaystyle \{X=k\},} { Y i } {\displaystyle \{Y_{i}\}} follows a multinomial distribution, { Y i } ∣ ( X = k ) ∼ M u l t i n o m ( k , p i ) , {\displaystyle
Poisson_distribution
Method for model fitting in statistics
S}{\partial \beta _{j}}}({\hat {\boldsymbol {\beta }}})=0} . The Gauss–Markov theorem shows that, when this is so, β ^ {\displaystyle {\hat {\boldsymbol {\beta
Weighted_least_squares
Type of statistical model
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Multilevel_model
Statistical model
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Fixed_effects_model
Statistical modeling method
regression and probit regression for binary data. Multinomial logistic regression and multinomial probit regression for categorical data. Ordered logit
Linear_regression
Statistical modeling technique
abnormal growth. The idea of estimating a median regression slope, a major theorem about minimizing sum of the absolute deviances and a geometrical algorithm
Quantile_regression
Random model in mathematics
_{i=1}^{k}a_{i}^{{\bar {n}}_{i}}}{(\sum _{i}a_{i})^{\bar {n}}}}} where we use the multinomial coefficient. Conditional on the urn ending up with ( a i + n i ) {\displaystyle
Pólya_urn_model
Random measure in probability theory
Y_{i}=nP_{n}(A_{i})} form a multinomial distribution with event probabilities P ( A i ) {\displaystyle P(A_{i})} The covariance matrix of this multinomial distribution
Empirical_measure
Equivalence class in mathematics
}{m_{1}!\cdots m_{k}!}}} is the multinomial coefficient. These two formulas follow directly from Pólya's enumeration theorem applied to the action of the
Necklace_(combinatorics)
Regression model for ordinal dependent variables
making no assumptions of the interval distances between options. Multinomial logit Multinomial probit McCullagh, Peter (1980). "Regression Models for Ordinal
Ordered_logit
Observation that in many real-life datasets, the leading digit is likely to be small
Accounting. 3 (3). Ostrovski, Vladimir (May 2017). "Testing equivalence of multinomial distributions". Statistics & Probability Letters. 124: 77–82. doi:10
Benford's_law
t-distribution. The negative multinomial distribution, a generalization of the negative binomial distribution. The Dirichlet negative multinomial distribution, a generalization
List of probability distributions
List_of_probability_distributions
Statistical estimation technique
When OLS is used on data with homoscedastic errors, the Gauss–Markov theorem applies, so the GLS estimate is the best linear unbiased estimator for
Generalized_least_squares
Least squares approximation of linear functions to data
and differentiation — this is an application of polynomial fitting. Multinomials in more than one independent variable, including surface fitting Curve
Linear_least_squares
Regression analysis for modeling ordinal data
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Ordinal_regression
Statistical method
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Partial least squares regression
Partial_least_squares_regression
Principle in Bayesian statistics
most probable result. The probability of any particular result is the multinomial distribution, P r ( p ) = W ⋅ m − N {\displaystyle Pr(\mathbf {p} )=W\cdot
Principle_of_maximum_entropy
Moving average and polynomial regression method for smoothing data
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Local_regression
Overview of and topical guide to machine learning
statistics Bayesian knowledge base Naive Bayes Gaussian Naive Bayes Multinomial Naive Bayes Averaged One-Dependence Estimators (AODE) Bayesian Belief
Outline_of_machine_learning
Method for solving certain optimization problems
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Iteratively reweighted least squares
Iteratively_reweighted_least_squares
Approximation method in statistics
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Non-linear_least_squares
Experiment methodology
determine which of the variants is more effective. Multivariate testing or multinomial testing is similar to A/B testing but may test more than two versions
A/B_testing
Probability distribution
recognized as Pascal's triangle. Mathematics portal Logistic regression Multinomial distribution Negative binomial distribution Beta-binomial distribution
Binomial_distribution
logistic regression. Multiclass classification methods include multinomial probit and multinomial logit. When the classification function is not perfect, false
Classification_rule
Scientific study of digital information
log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ2 test: mutual information can be considered
Information_theory
Approximation method in statistics
after reading Gauss's work, Laplace, after proving the central limit theorem, used it to give a large sample justification for the method of least squares
Least_squares
Generalized method of moments estimator in econometrics
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Arellano–Bond_estimator
Product of numbers from 1 to n
Dickson, Leonard E. (1919). "Chapter IX: Divisibility of factorials and multinomial coefficients". History of the Theory of Numbers. Vol. 1. Carnegie Institution
Factorial
Statistical model containing both fixed effects and random effects
{\boldsymbol {u}}} , respectively. This is a consequence of the Gauss–Markov theorem when the conditional variance of the outcome is not scalable to the identity
Mixed_model
(F:D) McCullagh's parametrization of the Cauchy distributions / (1:C) Multinomial distribution / (F:D) Multivariate normal distribution / Gau Negative
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Method of estimating the parameters of a statistical model, given observations
, … , x m {\displaystyle x_{1},\ x_{2},\ldots ,x_{m}} is called the multinomial and has the form: f ( x 1 , x 2 , … , x m ∣ p 1 , p 2 , … , p m ) = n
Maximum_likelihood_estimation
Concept in regression analysis mathematics
epsilon-insensitive loss leads to support vector regression. The representer theorem guarantees that the solution can be written as: f ( x ) = ∑ i = 1 n c i
Regularized_least_squares
restrictive assumption of mutually exclusive alternatives, which characterizes multinomial discrete choice methods. Ashford, J.R.; Sowden, R.R. (September 1970)
Multivariate_probit_model
Set of statistical processes for estimating the relationships among variables
variables. For categorical variables with more than two values there is the multinomial logit. For ordinal variables with more than two values, there are the
Regression_analysis
Mathematical function for the probability a given outcome occurs in an experiment
yes/no/maybe in a survey); a generalization of the Bernoulli distribution Multinomial distribution, for the number of each type of categorical outcome, given
Probability_distribution
Points with no three in a line
Gijswijt's upper bound. Jiang showed that by precisely examining the multinomial coefficients that come out of Ellenberg and Gijswijt's proof, one can
Cap_set
Concept in combinatorics (part of mathematics)
\end{bmatrix}}_{q}z^{n}.\end{aligned}}} One may further define the q-multinomial coefficients [ n k 1 , … , k m ] q = [ n ] ! q [ k 1 ] ! q ⋯ [ k m ]
Q-Pochhammer_symbol
Type of mathematical expression
called a trinomial. A polynomial with two or more terms is also called a multinomial. A real polynomial is a polynomial with real coefficients. When it is
Polynomial
Statistical test
G-test from the log-likelihood ratio test where the underlying model is a multinomial model. Suppose we had a sample O = ( O 1 , … , O m ) {\displaystyle O=(O_{1}
G-test
Kuder–Richardson Formula 20 Linear discriminant analysis Multinomial distribution Multinomial logit Multinomial probit Multiple correspondence analysis Odds ratio
List of analyses of categorical data
List_of_analyses_of_categorical_data
Concept in statistical mathematics
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Segmented_regression
Problem in machine learning and statistical classification
learning and statistical classification, multiclass classification or multinomial classification is the problem of classifying instances into one of three
Multiclass_classification
Principle in genetics
Hardy–Weinberg principle, also known as the Hardy–Weinberg equilibrium, model, theorem, or law, states that allele and genotype frequencies in a population will
Hardy–Weinberg_principle
Statistical model
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Random_effects_model
Statistical estimation method
detailed example, refer to: Tetsuo Yai, Seiji Iwakura, Shigeru Morichi, Multinomial probit with structured covariance for route choice behavior, Transportation
Binary_regression
Formula for the derivative of a product
derivative of an arbitrary number of factors, one has a similar formula with multinomial coefficients: ( ∏ i = 1 k f i ) ( n ) = ∑ j 1 + j 2 + ⋯ + j k = n ( n
Product_rule
Concept in statistical analysis
preferred brand of cereal, then probit or logit regression (or multinomial probit or multinomial logit) can be used. If both variables are ordinal, meaning
Bivariate_analysis
Constrained least squares problem
above, and an active set method called TNT-NN. M-matrix Perron–Frobenius theorem Chen, Donghui; Plemmons, Robert J. (2009). Nonnegativity constraints in
Non-negative_least_squares
Statistical model
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Mixed_logit
Two propositions or events that cannot both be true
charges, charges, and death sentences. In this case, the multinomial probit or multinomial logit technique is used. Contrariety Dichotomy Disjoint sets
Mutual_exclusivity
Statistical regression technique
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Multilevel regression with poststratification
Multilevel_regression_with_poststratification
Distinction between nominal, ordinal, interval and ratio variables
(specific blood type, political party, word, etc.) categorical multinomial logit, multinomial probit ordinal ordering categories or integer or real number
Level_of_measurement
Statistical method
bootstrap samples items from a list of size n with counts drawn from a multinomial distribution. If W i {\displaystyle W_{i}} denotes the number times element
Bootstrapping_(statistics)
Visualization method for regularization
regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
L-curve
Statistical regression where the dependent variable can take only two values
1935. Generalized linear model Limited dependent variable Logit model Multinomial probit Multivariate probit models Ordered probit and ordered logit model
Probit_model
Particular case of the generalized extreme value distribution
Gompertz function is obtained. In the latent variable formulation of the multinomial logit model — common in discrete choice theory — the errors of the latent
Gumbel_distribution
MULTINOMIAL THEOREM
MULTINOMIAL THEOREM
MULTINOMIAL THEOREM
MULTINOMIAL THEOREM
Boy/Male
Hindu, Indian
The Sun
Girl/Female
Hindu, Indian
Tender Leaves
Boy/Male
African, Hindu, Indian
Friends
Male
Finnish
Finnish myth name of the rival of Väinämöinen, possibly JOUKAHAINEN means "great, large."
Boy/Male
German
Eagle.
Boy/Male
Sikh
Of gods family
Boy/Male
Tamil
Pious
Boy/Male
Anglo Saxon
Gives.
Girl/Female
Muslim
Boy/Male
Muslim
Refined
MULTINOMIAL THEOREM
MULTINOMIAL THEOREM
MULTINOMIAL THEOREM
MULTINOMIAL THEOREM
MULTINOMIAL THEOREM
n. & a.
Same as Polynomial.
n.
A theorem or proposition so easy of demonstration as to be almost self-evident.
n.
The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.
n.
That which is considered and established as a principle; hence, sometimes, a rule.
a.
Theorematic.
a.
Alt. of Theorematical
v. t.
To formulate into a theorem.
a.
Alt. of Multinominous
n.
A statement of a principle to be demonstrated.
n.
One who constructs theorems.
n.
A numerical coefficient in any particular case of the binomial theorem.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
a.
Containing many names or terms; multinominal; as, the polynomial theorem.