# scipy.stats.dirichlet_multinomial#

scipy.stats.dirichlet_multinomial = <scipy.stats._multivariate.dirichlet_multinomial_gen object>[source]#

A Dirichlet multinomial random variable.

The Dirichlet multinomial distribution is a compound probability distribution: it is the multinomial distribution with number of trials n and class probabilities p randomly sampled from a Dirichlet distribution with concentration parameters alpha.

Parameters:
alphaarray_like

The concentration parameters. The number of entries along the last axis determines the dimensionality of the distribution. Each entry must be strictly positive.

nint or array_like

The number of trials. Each element must be a strictly positive integer.

seed{None, int, np.random.RandomState, np.random.Generator}, optional

Used for drawing random variates. If seed is None, the RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed. If seed is already a RandomState or Generator instance, then that object is used. Default is None.

scipy.stats.dirichlet

The dirichlet distribution.

scipy.stats.multinomial

The multinomial distribution.

References

[1]

Dirichlet-multinomial distribution, Wikipedia, https://www.wikipedia.org/wiki/Dirichlet-multinomial_distribution

Examples

>>> from scipy.stats import dirichlet_multinomial

Get the PMF

>>> n = 6  # number of trials
>>> alpha = [3, 4, 5]  # concentration parameters
>>> x = [1, 2, 3]  # counts
>>> dirichlet_multinomial.pmf(x, alpha, n)
0.08484162895927604

If the sum of category counts does not equal the number of trials, the probability mass is zero.

>>> dirichlet_multinomial.pmf(x, alpha, n=7)
0.0

Get the log of the PMF

>>> dirichlet_multinomial.logpmf(x, alpha, n)
-2.4669689491013327

Get the mean

>>> dirichlet_multinomial.mean(alpha, n)
array([1.5, 2. , 2.5])

Get the variance

>>> dirichlet_multinomial.var(alpha, n)
array([1.55769231, 1.84615385, 2.01923077])

Get the covariance

>>> dirichlet_multinomial.cov(alpha, n)
array([[ 1.55769231, -0.69230769, -0.86538462],
[-0.69230769,  1.84615385, -1.15384615],
[-0.86538462, -1.15384615,  2.01923077]])

Alternatively, the object may be called (as a function) to fix the alpha and n parameters, returning a “frozen” Dirichlet multinomial random variable.

>>> dm = dirichlet_multinomial(alpha, n)
>>> dm.pmf(x)
0.08484162895927579

All methods are fully vectorized. Each element of x and alpha is a vector (along the last axis), each element of n is an integer (scalar), and the result is computed element-wise.

>>> x = [[1, 2, 3], [4, 5, 6]]
>>> alpha = [[1, 2, 3], [4, 5, 6]]
>>> n = [6, 15]
>>> dirichlet_multinomial.pmf(x, alpha, n)
array([0.06493506, 0.02626937])
>>> dirichlet_multinomial.cov(alpha, n).shape  # both covariance matrices
(2, 3, 3)

Broadcasting according to standard NumPy conventions is supported. Here, we have four sets of concentration parameters (each a two element vector) for each of three numbers of trials (each a scalar).

>>> alpha = [[3, 4], [4, 5], [5, 6], [6, 7]]
>>> n = [[6], [7], [8]]
>>> dirichlet_multinomial.mean(alpha, n).shape
(3, 4, 2)

Methods

 logpmf(x, alpha, n): Log of the probability mass function. pmf(x, alpha, n): Probability mass function. mean(alpha, n): Mean of the Dirichlet multinomial distribution. var(alpha, n): Variance of the Dirichlet multinomial distribution. cov(alpha, n): The covariance of the Dirichlet multinomial distribution.