gamma_lpdf() [1/3]

template<bool propto, typename T_y_cl , typename T_shape_cl , typename T_inv_scale_cl , require_all_prim_or_rev_kernel_expression_t< T_y_cl, T_shape_cl, T_inv_scale_cl > * = nullptr, require_any_not_stan_scalar_t< T_y_cl, T_shape_cl, T_inv_scale_cl > * = nullptr>

return_type_t< T_y_cl, T_shape_cl, T_inv_scale_cl > stan::math::gamma_lpdf	(	const T_y_cl &	y,
		const T_shape_cl &	alpha,
		const T_inv_scale_cl &	beta
	)

The log of a gamma density for y with the specified shape and inverse scale parameters.

Shape and inverse scale parameters must be greater than 0. y must be greater than or equal to 0.

\begin{eqnarray*} y &\sim& \mbox{\sf{Gamma}}(\alpha, \beta) \\ \log (p (y \, |\, \alpha, \beta) ) &=& \log \left( \frac{\beta^\alpha}{\Gamma(\alpha)} y^{\alpha - 1} \exp^{- \beta y} \right) \\ &=& \alpha \log(\beta) - \log(\Gamma(\alpha)) + (\alpha - 1) \log(y) - \beta y\\ & & \mathrm{where} \; y > 0 \end{eqnarray*}

Template Parameters

T_y_cl	type of scalar
T_shape_cl	type of shape
T_inv_scale_cl	type of inverse scale

Parameters

y	A scalar variable.
alpha	Shape parameter.
beta	Inverse scale parameter.

Exceptions

std::domain_error	if alpha is not greater than 0.
std::domain_error	if beta is not greater than 0.
std::domain_error	if y is not greater than or equal to 0.

Definition at line 47 of file gamma_lpdf.hpp.