Multinomial

Parameters	${\displaystyle n>0}$ number of trials (integer) ${\displaystyle k>0}$ number of mutually exclusive events (integer) ${\displaystyle p_{1},\ldots ,p_{k))$ event probabilities, where ${\displaystyle p_{1}+\dots +p_{k}=1}$
Support	${\displaystyle \left\lbrace (x_{1},\dots ,x_{k})\,{\Big \vert }\,\sum _{i=1}^{k}x_{i}=n,x_{i}\geq 0\ (i=1,\dots ,k)\right\rbrace }$
PMF	${\displaystyle {\frac {n!}{x_{1}!\cdots x_{k}!))p_{1}^{x_{1))\cdots p_{k}^{x_{k))}$
Mean	${\displaystyle \operatorname {E} (X_{i})=np_{i))$
Variance	${\displaystyle \operatorname {Var} (X_{i})=np_{i}(1-p_{i})}$ ${\displaystyle \operatorname {Cov} (X_{i},X_{j})=-np_{i}p_{j}~~(i\neq j)}$
Entropy	${\displaystyle -\log(n!)-n\sum _{i=1}^{k}p_{i}\log(p_{i})+\sum _{i=1}^{k}\sum _{x_{i}=0}^{n}{\binom {n}{x_{i))}p_{i}^{x_{i))(1-p_{i})^{n-x_{i))\log(x_{i}!)}$
MGF	${\displaystyle {\biggl (}\sum _{i=1}^{k}p_{i}e^{t_{i)){\biggr )}^{n))$
CF	${\displaystyle \left(\sum _{j=1}^{k}p_{j}e^{it_{j))\right)^{n))$ where ${\displaystyle i^{2}=-1}$
PGF	${\displaystyle {\biggl (}\sum _{i=1}^{k}p_{i}z_{i}{\biggr )}^{n}{\text{ for ))(z_{1},\ldots ,z_{k})\in \mathbb {C} ^{k))$

In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each side of a k-sided die rolled n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

When k is 2 and n is 1, the multinomial distribution is the Bernoulli distribution. When k is 2 and n is bigger than 1, it is the binomial distribution. When k is bigger than 2 and n is 1, it is the categorical distribution. The term "multinoulli" is sometimes used for the categorical distribution to emphasize this four-way relationship (so n determines the suffix, and k the prefix).

The Bernoulli distribution models the outcome of a single Bernoulli trial. In other words, it models whether flipping a (possibly biased) coin one time will result in either a success (obtaining a head) or failure (obtaining a tail). The binomial distribution generalizes this to the number of heads from performing n independent flips (Bernoulli trials) of the same coin. The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a categorical distribution, such as rolling a k-sided die n times.

Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p₁, ..., p_k, and n independent trials. Since the k outcomes are mutually exclusive and one must occur we have p_i ≥ 0 for i = 1, ..., k and ${\displaystyle \sum _{i=1}^{k}p_{i}=1}$. Then if the random variables X_i indicate the number of times outcome number i is observed over the n trials, the vector X = (X₁, ..., X_k) follows a multinomial distribution with parameters n and p, where p = (p₁, ..., p_k). While the trials are independent, their outcomes X_i are dependent because they must be summed to n.

Definitions

Probability mass function

Suppose one does an experiment of extracting n balls of k different colors from a bag, replacing the extracted balls after each draw. Balls of the same color are equivalent. Denote the variable which is the number of extracted balls of color i (i = 1, ..., k) as X_i, and denote as p_i the probability that a given extraction will be in color i. The probability mass function of this multinomial distribution is:

${\displaystyle {\begin{aligned}f(x_{1},\ldots ,x_{k};n,p_{1},\ldots ,p_{k})&{}=\Pr(X_{1}=x_{1}{\text{ and ))\dots {\text{ and ))X_{k}=x_{k})\\&{}={\begin{cases}{\displaystyle {n! \over x_{1}!\cdots x_{k}!}p_{1}^{x_{1))\times \cdots \times p_{k}^{x_{k))},\quad &{\text{when ))\sum _{i=1}^{k}x_{i}=n\\\\0&{\text{otherwise,))\end{cases))\end{aligned))}$

for non-negative integers x₁, ..., x_k.

The probability mass function can be expressed using the gamma function as:

${\displaystyle f(x_{1},\dots ,x_{k};p_{1},\ldots ,p_{k})={\frac {\Gamma (\sum _{i}x_{i}+1)}{\prod _{i}\Gamma (x_{i}+1)))\prod _{i=1}^{k}p_{i}^{x_{i)).}$

This form shows its resemblance to the Dirichlet distribution, which is its conjugate prior.

Example

Suppose that in a three-way election for a large country, candidate A received 20% of the votes, candidate B received 30% of the votes, and candidate C received 50% of the votes. If six voters are selected randomly, what is the probability that there will be exactly one supporter for candidate A, two supporters for candidate B and three supporters for candidate C in the sample?

Note: Since we’re assuming that the voting population is large, it is reasonable and permissible to think of the probabilities as unchanging once a voter is selected for the sample. Technically speaking this is sampling without replacement, so the correct distribution is the multivariate hypergeometric distribution, but the distributions converge as the population grows large in comparison to a fixed sample size^[1].

${\displaystyle \Pr(A=1,B=2,C=3)={\frac {6!}{1!2!3!))(0.2^{1})(0.3^{2})(0.5^{3})=0.135}$

Properties

Expected value and variance

The expected number of times the outcome i was observed over n trials is

${\displaystyle \operatorname {E} (X_{i})=np_{i}.\,}$

The covariance matrix is as follows. Each diagonal entry is the variance of a binomially distributed random variable, and is therefore

${\displaystyle \operatorname {Var} (X_{i})=np_{i}(1-p_{i}).\,}$

The off-diagonal entries are the covariances:

${\displaystyle \operatorname {Cov} (X_{i},X_{j})=-np_{i}p_{j}\,}$

for i, j distinct.

All covariances are negative because for fixed n, an increase in one component of a multinomial vector requires a decrease in another component.

When these expressions are combined into a matrix with i, j element ${\displaystyle \operatorname {cov} (X_{i},X_{j}),}$ the result is a k × k positive-semidefinite covariance matrix of rank k − 1. In the special case where k = n and where the p_i are all equal, the covariance matrix is the centering matrix.

The entries of the corresponding correlation matrix are

${\displaystyle \rho (X_{i},X_{i})=1.}$

${\displaystyle \rho (X_{i},X_{j})={\frac {\operatorname {Cov} (X_{i},X_{j})}{\sqrt {\operatorname {Var} (X_{i})\operatorname {Var} (X_{j})))}={\frac {-p_{i}p_{j)){\sqrt {p_{i}(1-p_{i})p_{j}(1-p_{j})))}=-{\sqrt {\frac {p_{i}p_{j)){(1-p_{i})(1-p_{j})))}.}$

Note that the number of trials n drops out of this expression.

Each of the k components separately has a binomial distribution with parameters n and p_i, for the appropriate value of the subscript i.

The support of the multinomial distribution is the set

${\displaystyle \{(n_{1},\dots ,n_{k})\in \mathbb {N} ^{k}\mid n_{1}+\cdots +n_{k}=n\}.\,}$

Its number of elements is

${\displaystyle {n+k-1 \choose k-1}.}$

Matrix notation

In matrix notation,

${\displaystyle \operatorname {E} (\mathbf {X} )=n\mathbf {p} ,\,}$

and

${\displaystyle \operatorname {Var} (\mathbf {X} )=n\lbrace \operatorname {diag} (\mathbf {p} )-\mathbf {p} \mathbf {p} ^{\rm {T))\rbrace ,\,}$

with p^T = the row vector transpose of the column vector p.

Visualization

As slices of generalized Pascal's triangle

Just like one can interpret the binomial distribution as (normalized) one-dimensional (1D) slices of Pascal's triangle, so too can one interpret the multinomial distribution as 2D (triangular) slices of Pascal's pyramid, or 3D/4D/+ (pyramid-shaped) slices of higher-dimensional analogs of Pascal's triangle. This reveals an interpretation of the range of the distribution: discretized equilateral "pyramids" in arbitrary dimension—i.e. a simplex with a grid.^{[citation needed]}

As polynomial coefficients

Similarly, just like one can interpret the binomial distribution as the polynomial coefficients of ${\displaystyle (p+q)^{n))$ when expanded, one can interpret the multinomial distribution as the coefficients of ${\displaystyle (p_{1}+p_{2}+p_{3}+\cdots +p_{k})^{n))$ when expanded, noting that just the coefficients must sum up to 1.

Large deviation theory

Asymptotics

By Stirling's formula, at the limit of ${\displaystyle N,x_{1},...,x_{n}\to \infty }$, we have

$${\displaystyle \ln {\binom {N}{x_{1},\cdots x_{n))}p_{1}^{x_{1))\cdots p_{n}^{x_{n))=-ND_{KL}({\hat {p))\|p)-{\frac {1}{2))\sum _{i}\ln({\hat {p))_{i})-{\frac {n-1}{2))\ln(2\pi )+o(1)}$$

${\displaystyle {\hat {p))_{i}=x_{i}/N}$

${\displaystyle D_{KL))$

This formula can be interpreted as follows.

Consider ${\displaystyle \Delta _{n))$, the space of all possible distributions over the categories ${\displaystyle \{1,2,...,n\))$. It is a simplex. After ${\displaystyle N}$ independent samples from the categorical distribution ${\displaystyle p}$ (which is how we construct the multinomial distribution), we obtain an empirical distribution ${\displaystyle {\hat {p))}$.

By the asymptotic formula, the probability that empirical distribution ${\displaystyle {\hat {p))}$deviates from the actual distribution ${\displaystyle p}$ decays exponentially, at a rate ${\displaystyle ND_{KL}({\hat {p))\|p)}$. The more experiments and the more different ${\displaystyle {\hat {p))}$ is from ${\displaystyle p}$, the less likely it is to see such an empirical distribution.

If ${\displaystyle A}$ is a closed subset of ${\displaystyle \Delta _{n))$, then by dividing up ${\displaystyle A}$ into pieces, and reasoning about the growth rate of ${\displaystyle Pr({\hat {p))\in A_{\epsilon })}$ on each piece ${\displaystyle A_{\epsilon ))$, we obtain Sanov's theorem, which states that

$${\displaystyle \lim _{N\to \infty }{\frac {1}{N))\ln Pr({\hat {p))\in A)=-\inf _((\hat {p))\in A}D_{KL}({\hat {p))\|p)}$$

Concentration at large N

Due to the exponential decay, at large ${\displaystyle N}$, almost all the probability mass is concentrated in a small neighborhood of ${\displaystyle p}$. In this small neighborhood, we can take the first nonzero term in the Taylor expansion of ${\displaystyle D_{KL))$, to obtain

$${\displaystyle \ln {\binom {N}{x_{1},\cdots x_{n))}p_{1}^{x_{1))\cdots p_{n}^{x_{n))\approx -{\frac {N}{2))\sum _{i}{\frac {({\hat {p))_{i}-p_{i})^{2)){p_{i))}=-{\frac {1}{2))\sum _{i}{\frac {(x_{i}-Np_{i})^{2)){Np_{i))))$$

Theorem. At the ${\displaystyle N\to \infty }$ limit, ${\displaystyle N\sum _{i}{\frac {({\hat {p))_{i}-p_{i})^{2)){p_{i))}=\sum _{i}{\frac {(x_{i}-Np_{i})^{2)){Np_{i))))$ converges in distribution to the chi-squared distribution ${\displaystyle \chi ^{2}(n-1)}$.

Proof. The space of all distributions over categories ${\displaystyle \{1,2,\ldots ,n\))$ is a simplex: ${\displaystyle \Delta _{n}=\left\{(y_{1},\ldots ,y_{n})\colon y_{1},\ldots ,y_{n}\geq 0,\sum _{i}y_{i}=1\right\))$, and the set of all possible empirical distributions after ${\displaystyle N}$ experiments is a subset of the simplex: ${\displaystyle \Delta _{n,N}=\left\{(x_{1}/N,\ldots ,x_{n}/N)\colon x_{1},\ldots ,x_{n}\in \mathbb {N} ,\sum _{i}x_{i}=N\right\))$. That is, it is the intersection between ${\displaystyle \Delta _{n))$ and the lattice ${\displaystyle (\mathbb {Z} ^{n})/N}$.

As ${\displaystyle N}$ increases, most of the probability mass is concentrated in a subset of ${\displaystyle \Delta _{n,N))$ near ${\displaystyle p}$, and the probability distribution near ${\displaystyle p}$ becomes well-approximated by

$${\displaystyle {\binom {N}{x_{1},\cdots x_{n))}p_{1}^{x_{1))\cdots p_{n}^{x_{n))\approx e^{-{\frac {N}{2))\sum _{i}{\frac {({\hat {p))_{i}-p_{i})^{2)){p_{i))))}$$

${\displaystyle 1/{\sqrt {N))}$

${\displaystyle 1/N}$

${\displaystyle N}$

${\displaystyle \Delta _{n,N))$

${\displaystyle \Delta _{n))$

${\displaystyle \rho ({\hat {p)))=Ce^{-{\frac {N}{2))\sum _{i}{\frac {({\hat {p))_{i}-p_{i})^{2)){p_{i))))}$

${\displaystyle C}$

Finally, since the simplex ${\displaystyle \Delta _{n))$ is not all of ${\displaystyle \mathbb {R} ^{n))$, but only within a ${\displaystyle (n-1)}$-dimensional plane, we obtain the desired result.

Conditional concentration at large N

The above concentration phenomenon can be easily generalized to the case where we condition upon linear constraints. This is the theoretical justification for Pearson's chi-squared test.

Theorem. If we impose ${\displaystyle k+1}$ independent linear constraints

$${\displaystyle {\begin{cases}\sum _{i}{\hat {p))_{i}=1,\\\sum _{i}a_{1i}{\hat {p))_{i}=b_{1},\\\sum _{i}a_{2i}{\hat {p))_{i}=b_{2},\\\cdots ,\\\sum _{i}a_{ki}{\hat {p))_{i}=b_{k}\end{cases))}$$

${\displaystyle p}$

${\displaystyle N\to \infty }$

${\displaystyle N\sum _{i}{\frac {({\hat {p))_{i}-p_{i})^{2)){p_{i))}=\sum _{i}{\frac {(x_{i}-Np_{i})^{2)){Np_{i))))$

${\displaystyle \chi ^{2}(n-1-k)}$

Proof. The same proof applies, but this time ${\displaystyle \Delta _{n,N))$ is the intersection of ${\displaystyle (\mathbb {Z} ^{n})/N}$ with ${\displaystyle \Delta _{n))$ and ${\displaystyle k}$ hyperplanes, all linearly independent, so the probability density ${\displaystyle \rho ({\hat {p)))}$ is restricted to a ${\displaystyle (n-k-1)}$-dimensional plane.

The above theorem is not entirely satisfactory, because by definition, every one of ${\displaystyle {\hat {p))_{1},{\hat {p))_{2},...,{\hat {p))_{n))$ must be a rational number, whereas ${\displaystyle p_{1},p_{2},...,p_{n))$ may be chosen from any number in ${\displaystyle [0,1]}$. In particular, ${\displaystyle p}$ may satisfy linear constraints that no ${\displaystyle {\hat {p))}$ can possibly satisfy, such as ${\displaystyle {\sqrt {2))\;p_{1}=1}$. The next theorem fixes this issue:

Theorem.

• Given functions ${\displaystyle f_{1},...,f_{k))$, such that they are continuously differentiable in a neighborhood of ${\displaystyle p}$, and the vectors ${\displaystyle (1,1,...,1),\nabla f_{1}(p),...,\nabla f_{k}(p)}$ are linearly independent;
• given sequences ${\displaystyle \epsilon _{1}(N),...,\epsilon _{n}(N)}$, such that asymptotically ${\displaystyle {\frac {1}{N))\ll \epsilon _{k}(N)\ll {\frac {1}{\sqrt {N))))$ for each ${\displaystyle k\in \{1,...,n\))$;
• then for the multinomial distribution conditional on constraints ${\displaystyle f_{1}({\hat {p)))\in [f_{1}(p)-\epsilon _{1}(N),f_{1}(p)+\epsilon _{1}(N)],...,f_{n}({\hat {p)))\in [f_{n}(p)-\epsilon _{n}(N),f_{n}(p)+\epsilon _{n}(N)]}$, we have the quantity ${\displaystyle N\sum _{i}{\frac {({\hat {p))_{i}-p_{i})^{2)){p_{i))}=\sum _{i}{\frac {(x_{i}-Np_{i})^{2)){Np_{i))))$ converging in distribution to ${\displaystyle \chi ^{2}(n-1-k)}$ at the ${\displaystyle N\to \infty }$ limit.

In the case that all ${\displaystyle {\hat {p))_{i))$ are equal, the Theorem reduces to the concentration of entropies around the Maximum Entropy.^[2][3]

Related distributions

In some fields such as natural language processing, categorical and multinomial distributions are synonymous 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 ${\displaystyle 1\dots K}$; in this form, a categorical distribution is equivalent to a multinomial distribution over a single trial.

• When k = 2, the multinomial distribution is the binomial distribution.
• Categorical distribution, the distribution of each trial; for k = 2, this is the Bernoulli distribution.
• The Dirichlet distribution is the conjugate prior of the multinomial in Bayesian statistics.
• Dirichlet-multinomial distribution.
• Beta-binomial distribution.
• Negative multinomial distribution
• Hardy–Weinberg principle (it is a trinomial distribution with probabilities ${\displaystyle (\theta ^{2},2\theta (1-\theta ),(1-\theta )^{2})}$)

Statistical inference

Equivalence tests for multinomial distributions

The goal of equivalence testing is to establish the agreement between a theoretical multinomial distribution and observed counting frequencies. The theoretical distribution may be a fully specified multinomial distribution or a parametric family of multinomial distributions.

Let ${\displaystyle q}$ denote a theoretical multinomial distribution and let ${\displaystyle p}$ be a true underlying distribution. The distributions ${\displaystyle p}$ and ${\displaystyle q}$ are considered equivalent if ${\displaystyle d(p,q)<\varepsilon }$ for a distance ${\displaystyle d}$ and a tolerance parameter ${\displaystyle \varepsilon >0}$. The equivalence test problem is ${\displaystyle H_{0}=\{d(p,q)\geq \varepsilon \))$ versus ${\displaystyle H_{1}=\{d(p,q)<\varepsilon \))$. The true underlying distribution ${\displaystyle p}$ is unknown. Instead, the counting frequencies ${\displaystyle p_{n))$ are observed, where ${\displaystyle n}$ is a sample size. An equivalence test uses ${\displaystyle p_{n))$ to reject ${\displaystyle H_{0))$. If ${\displaystyle H_{0))$ can be rejected then the equivalence between ${\displaystyle p}$ and ${\displaystyle q}$ is shown at a given significance level. The equivalence test for Euclidean distance can be found in text book of Wellek (2010).^[4] The equivalence test for the total variation distance is developed in Ostrovski (2017).^[5] The exact equivalence test for the specific cumulative distance is proposed in Frey (2009).^[6]

The distance between the true underlying distribution ${\displaystyle p}$ and a family of the multinomial distributions ${\displaystyle {\mathcal {M))}$ is defined by ${\displaystyle d(p,{\mathcal {M)))=\min _{h\in {\mathcal {M))}d(p,h)}$. Then the equivalence test problem is given by ${\displaystyle H_{0}=\{d(p,{\mathcal {M)))\geq \varepsilon \))$ and ${\displaystyle H_{1}=\{d(p,{\mathcal {M)))<\varepsilon \))$. The distance ${\displaystyle d(p,{\mathcal {M)))}$ is usually computed using numerical optimization. The tests for this case are developed recently in Ostrovski (2018).^[7]

Random variate generation

First, reorder the parameters ${\displaystyle p_{1},\ldots ,p_{k))$ such that they are sorted in descending order (this is only to speed up computation and not strictly necessary). Now, for each trial, draw an auxiliary variable X from a uniform (0, 1) distribution. The resulting outcome is the component

${\displaystyle j=\min \left\{j'\in \{1,\dots ,k\}\colon \left(\sum _{i=1}^{j'}p_{i}\right)-X\geq 0\right\}.}$

{X_j = 1, X_k = 0 for k ≠ j } is one observation from the multinomial distribution with ${\displaystyle p_{1},\ldots ,p_{k))$ and n = 1. A sum of independent repetitions of this experiment is an observation from a multinomial distribution with n equal to the number of such repetitions.

Sampling using repeated conditional binomial samples

Given the parameters ${\displaystyle p_{1},p_{2},\ldots ,p_{k))$ and a total for the sample ${\displaystyle N}$ such that ${\displaystyle \sum _{i=1}^{k}X_{i}=N}$, it is possible to sample sequentially for the number in an arbitrary state ${\displaystyle X_{i))$, by partitioning the state space into ${\displaystyle i}$ and not-${\displaystyle i}$, conditioned on any prior samples already taken, repeatedly.

Algorithm: Sequential conditional binomial sampling

S = N
rho = 1
for i in [1,k-1]:
    if rho != 0:
        X[i] ~ Binom(S,p[i]/rho)
    else X[i] = 0
    S = S - X[i]
    rho = rho - p[i]
X[k] = S

Heuristically, each application of the binomial sample reduces the available number to sample from and the conditional probabilities are likewise updated to ensure logical consistency.^[8]