dirichlet

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dirichlet#

class brainstate.random.dirichlet(alpha, size=None, key=None, dtype=None)#

Draw samples from the Dirichlet distribution.

Draw size samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference.

Parameters:

  • alpha (sequence of floats, length k) – Parameter of the distribution (length k for sample of length k).

  • size (int | Sequence[int] | integer | Sequence[integer] | None) – Output shape. If the given shape is, e.g., (m, n), then m * n * k samples are drawn. Default is None, in which case a vector of length k is returned.

  • key (int | Array | ndarray | None) – The key for the random number generator. If not given, the default random number generator is used.

Returns:

samples – The drawn samples, of shape (size, k).

Return type:

ndarray,

Raises:

ValueError – If any value in alpha is less than or equal to zero

Notes

The Dirichlet distribution is a distribution over vectors \(x\) that fulfil the conditions \(x_i>0\) and \(\sum_{i=1}^k x_i = 1\).

The probability density function \(p\) of a Dirichlet-distributed random vector \(X\) is proportional to

\[p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},\]

where \(\alpha\) is a vector containing the positive concentration parameters.

The method uses the following property for computation: let \(Y\) be a random vector which has components that follow a standard gamma distribution, then \(X = \frac{1}{\sum_{i=1}^k{Y_i}} Y\) is Dirichlet-distributed

References

Examples

Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces.

>>> import brainstate
>>> s = brainstate.random.dirichlet((10, 5, 3), 20).transpose()

>>> import matplotlib.pyplot as plt  # noqa
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")
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