# Multinomial Distribution

In probability theory, the 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. In a multinomial distribution, the analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number k of possible outcomes, with probabilities p1, ..., pk (so that pi ≥ 0 for i = 1, ..., k and ), and there are n independent trials. Then let the random variables Xi indicate the number of times outcome number i was observed over the n trials. The vector X = (X1, ..., Xk) follows a multinomial distribution with parameters n and p, where p = (p1, ..., pk).

Note that, in some fields, such as natural language processing, the categorical and multinomial distributions are conflated, and it is common to speak of a "multinomial distribution" when a categorical distribution is actually meant. This stems from the fact that it is sometimes convenient to express the outcome of a categorical distribution as a "1-of-K" vector (a vector with one element containing a 1 and all other elements containing a 0) rather than as an integer in the range ; in this form, a categorical distribution is equivalent to a multinomial distribution over a single observation.

Read more about Multinomial Distribution:  Properties, Example, Sampling From A Multinomial Distribution, To Simulate A Multinomial Distribution, Related Distributions

### Other articles related to "multinomial distribution, distribution":

Dirichlet-multinomial Distribution - Probability Mass Function - For A Multinomial Distribution Over Category Counts
... For a random vector of category counts, distributed according to a multinomial distribution, the marginal distribution is obtained by integrating out p which results in the following explicit formula ... in having an extra term at the front that looks like the factor at the front of a multinomial distribution ... Another form for this same compound distribution, written more compactly in terms of the beta function, B, is as follows ...
Variational Bayesian Methods - A More Complex Example
... Gaussian mixture model described as follows Note SymDir is the symmetric Dirichlet distribution of dimension, with the hyperparameter for each component ... The Dirichlet distribution is the conjugate prior of the categorical distribution or multinomial distribution ... is the Wishart distribution, which is the conjugate prior of the precision matrix (inverse covariance matrix) for a multivariate Gaussian distribution ...
Multinomial Distribution - Related Distributions
... When k = 2, the multinomial distribution is the binomial distribution ... The continuous analogue is Multivariate normal distribution ... Categorical distribution, the distribution of each trial for k = 2, this is the Bernoulli distribution ...
Dirichlet-multinomial Distribution
... and statistics, the Dirichlet-multinomial distribution is a probability distribution for a multivariate discrete random variable ... It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution, which is named after George Pólya) ... It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector, and a set of discrete samples is drawn from the categorical distribution ...
Categorical Distribution
... A categorical distribution is a discrete probability distribution whose sample space is the set of k individually identified items ... It is the generalization of the Bernoulli distribution for a categorical random variable ... In one formulation of the distribution, the sample space is taken to be a finite sequence of integers ...

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