dirichlet_multinomial_lpmf() [1/2]

template<bool propto, typename T_prior_size , require_eigen_col_vector_t< T_prior_size > * = nullptr>

return_type_t< T_prior_size > stan::math::dirichlet_multinomial_lpmf	(	const std::vector< int > &	ns,
		const T_prior_size &	alpha
	)

The log of the Dirichlet-Multinomial probability for the given integer vector $n$ and a vector of prior sample sizes, $\alpha$.

Each element of $\alpha$ must be greater than 0. Each element of $n$ must be greater than or equal to 0.

Suppose that $n = (n_1, \ldots, n_K)$ has a Dirichlet-multinomial distribution with prior sample size $\alpha = (\alpha_1,\ldots,\alpha_K)$ (also called the intensity):

$$(n_1,\ldots,n_K)\sim\mbox{DirMult}(\alpha_1,\ldots,\alpha_K)$$

Write $N = n_1 + \cdots + n_K$ and $\alpha_0 = \alpha_1 + \cdots + \alpha_K$, then the log probability mass function is given by

$$\log(p(n_1, \ldots, n_K\,|\,\alpha_1,\ldots,\alpha_K))=\log\left( \frac{\Gamma(\alpha_0)\Gamma(N+1)}{\Gamma(\alpha_0 + N)} \prod_{k=1}^K \frac{\Gamma(n_k + \alpha_k)}{\Gamma(\alpha_k) \Gamma(n_k+1)} \right)\\ = \log(N) + \log(B(\alpha_0, N)) - \sum_{k : n_k > 0} \bigl(\log(n_k) + \log(B(\alpha_k, n_k))\bigr)$$

The second identity is only valid for $N > 0$. For $N=0$, we have $\log(p(n\,|\,\alpha)) = 0$.

Template Parameters

T_prior_size

type of prior sample sizes

Parameters

ns	A vector of integers.
alpha	Prior sample sizes (or intensity vector).

Returns

The log of the Dirichlet-Multinomial probability.

Exceptions

std::domain_error	if any element of alpha is less than or equal to 0, or infinite.
std::domain_error	any element of ns is less than 0.

Definition at line 60 of file dirichlet_multinomial_lpmf.hpp.