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Information about Categorical Distribution

A categorical distribution is the most general distribution whose sample space is the set {1, 2, ..., n}.

It is the generalization of the Bernoulli distribution for a categorical random variable.

It should not be confused with the multinomial distribution.

Consider the extended analogy:
Bernoulli random variable : Bernoulli distribution : Binomial distribution :: Categorical random variable : Categorical distribution : Multinomial distribution

The distribution is completely given by the probabilities associated with each number k: , k = 1,...,n.

In the BUGS software, this distribution is represented as: 
>
BUGS: r ~ dcat(p[])
// BUGS assigns categories the values 1, 2, ..., K


Sufficient statistics from n independent observations are number of trials and count (or, equivalently, proportion) of observations in each category.

Conjugate prior is the Dirichlet distribution.

Probability distributions [  •  • [ edit] ]
Univariate Multivariate
Discrete: Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-MandelbrotEwens • multinomial • multivariate Polya
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lvy • Lvy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambdaDirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: bimodal • Cantor • conditional • equilibrium • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular
In probability theory, the sample space or universal sample space or event space, often denoted S, Ω or U (for "universe"), of an experiment or random trial is the set of all possible outcomes.
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Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability .
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The level of measurement of a variable in mathematics and statistics is a classification that was proposed in order to describe the nature of information contained within numbers assigned to objects and, therefore, within the variable.
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multinomial distribution is a generalization of the binomial distribution.

The binomial distribution is the probability distribution of the number of "successes" in n independent Bernoulli trials, with the same probability of "success" on each trial.
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OpenBUGS is a computer software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods. OpenBUGS is the open source variant of WinBUGS (Bayesian inference Using Gibbs Sampling).
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Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet), often denoted Dir(α), is a family of continuous multivariate probability distributions parametrized by the vector α of positive reals.
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probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.
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In statistics, in univariate data, each data point has only one scalar component. Or, when the statistical technique to be used, it contains only one dependent variable. The more general case is multivariate.
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A multivariate random variable or random vector is a vector X = (X1, ..., Xn) whose components are scalar-valued random variables on the same probability space (Ω, P).
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Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability .
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binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.
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Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each respectively possess energy Ei:


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In probability theory, a compound Poisson distribution is the probability distribution of a "Poisson-distributed number" of independent identically-distributed random variables. More precisely, suppose



i.e.
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degenerate distribution is the probability distribution of a discrete random variable whose support consists of only one value. Examples include a two-headed coin and rolling a die whose sides all show the same number.
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Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. The distribution is named after Carl Friedrich Gauss, who first conjectured and studied the distribution around 1800,
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geometric distribution is either of two discrete probability distributions:
  • the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set , or
  • the probability distribution of the number Y

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hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement.
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logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution derived from the Maclaurin series expansion



From this we obtain the identity


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negative binomial distribution is a discrete probability distribution. The Pascal distribution and the Polya distribution are special cases of the negative binomial.
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In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship (see external link below).
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Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event.
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Rademacher distribution, named after Hans Rademacher is a discrete probability distribution which has a 50% chance for either 1 or -1. The probability mass function of this distribution is



The Rademacher distribution has been used in bootstrapping.
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Skellam distribution is the discrete probability distribution of the difference of two correlated or uncorrelated random variables and having Poisson distributions with different expected values and .
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discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable.
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Yule-Simon distribution is a discrete probability distribution named after Udny Yule and Herbert Simon. Simon originally called it the Yule distribution.

The probability mass function of the Yule-Simon(ρ) distribution is


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zeta distribution is a discrete probability distribution. If X is a zeta-distributed random variable with parameter s, then the probability that X takes the integer value k is given by the probability mass function


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Zipf's law, publicized by Harvard linguist George Kingsley Zipf (IPA [zɪf]), stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency
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Zipf-Mandelbrot law is a discrete probability distribution. Also known as the Pareto-Zipf law, it is a power-law distribution on ranked data, named after the Harvard linguistics professor George Kingsley Zipf (1902-1950) who suggested a simpler distribution called Zipf's law, and
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In population genetics, Ewens's sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability
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multinomial distribution is a generalization of the binomial distribution.

The binomial distribution is the probability distribution of the number of "successes" in n independent Bernoulli trials, with the same probability of "success" on each trial.
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