Multinomial distribution


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.
The Bernoulli distribution models the outcome of a single Bernoulli trial. In other words, it models whether flipping a coin one time will result in either a success or failure. The binomial distribution generalizes this to the number of heads from performing n independent flips 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,..., pk, and n independent trials. Since the k outcomes are mutually exclusive and one must occur we have pi ≥ 0 for i = 1, ..., k and. Then if the random variables Xi indicate the number of times outcome number i is observed over the n trials, the vector X = follows a multinomial distribution with parameters n and p, where p = . While the trials are independent, their outcomes Xi are dependent because they must sum 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 as Xi, and denote as pi the probability that a given extraction will be in color i. The probability mass function of this multinomial distribution is:
for non-negative integers x1,..., xk.
The probability mass function can be expressed using the gamma function as:
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.

Properties

Normalization

The multinomial distribution is normalized according to:
where the sum is over all permutations of such that.

Expected value and variance

The expected number of times the outcome i was observed over n trials is
The covariance matrix is as follows. Each diagonal entry is the variance of a binomially distributed random variable, and is therefore
The off-diagonal entries are the covariances:
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 the result is a k × k positive-semidefinite covariance matrix of rank k − 1. In the special case where k = n and where the pi are all equal, the covariance matrix is the centering matrix.
The entries of the corresponding correlation matrix are
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 pi, for the appropriate value of the subscript i.
The support of the multinomial distribution is the set
Its number of elements is

Matrix notation

In matrix notation,
and
with = the row vector transpose of the column vector.

Visualization

As slices of generalized Pascal's triangle

Just like one can interpret the binomial distribution as one-dimensional slices of Pascal's triangle, so too can one interpret the multinomial distribution as 2D slices of Pascal's pyramid, or 3D/4D/+ 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.

As polynomial coefficients

Similarly, just like one can interpret the binomial distribution as the polynomial coefficients of when expanded, one can interpret the multinomial distribution as the coefficients of when expanded, noting that just the coefficients must sum up to 1.

Large deviation theory

Asymptotics

By Stirling's formula, at the limit of, we havewhere relative frequencies in the data can be interpreted as probabilities from the empirical distribution, and is the Kullback–Leibler divergence.
This formula can be interpreted as follows.
Consider, the space of all possible distributions over the categories. It is a simplex. After independent samples from the categorical distribution , we obtain an empirical distribution.
By the asymptotic formula, the probability that empirical distribution deviates from the actual distribution decays exponentially as we sample more data, at a rate of. The more experiments and the more different is from, the less likely it is to see such an empirical distribution.
If is a closed subset of, then by dividing up into pieces, and reasoning about the growth rate of on each piece, we obtain Sanov's theorem, which states that

Concentration at large ''n''

Due to the exponential decay, at large, almost all the probability mass is concentrated in a small neighborhood of. In this small neighborhood, we can take the first nonzero term in the Taylor expansion of, to obtain
This resembles the Gaussian distribution, which suggests the following theorem:
Theorem. At the limit, converges in distribution to the chi-squared distribution.
The space of all distributions over categories is a simplex:, and the set of all possible empirical distributions after experiments is a subset of the simplex:. That is, it is the intersection between and the lattice.
As increases, most of the probability mass is concentrated in a subset of near, and the probability distribution near becomes well-approximated by From this, we see that the subset upon which the mass is concentrated has radius on the order of, but the points in the subset are separated by distance on the order of, so at large, the points merge into a continuum.
To convert this from a discrete probability distribution to a continuous probability density, we need to multiply by the volume occupied by each point of in. However, by symmetry, every point occupies exactly the same volume, so we obtain a probability density, where is a constant.
Finally, since the simplex is not all of, but only within a -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 independent constraints. This is the theoretical justification for Pearson's chi-squared test.
Theorem.
  • Given functions, such that they are continuously differentiable in a neighborhood of, and the vectors are linearly independent;
  • given sequences, such that asymptotically for each ;
  • then for the multinomial distribution conditional on constraints, we have the quantity converging in distribution to at the limit.
In the case that all are equal, this reduces to the concentration of entropies around the maximum entropy.
This theorem can be shown by starting with the previous case, then taking the conditional on the constraints.

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 rather than as an integer in the range ; in this form, a categorical distribution is equivalent to a multinomial distribution over a single trial.