lbeta()

double stan::math::opencl_kernels::lbeta	(	double	a,
		double	b
	)

Return the log of the beta function applied to the specified arguments.

The beta function is defined for \(a > 0\) and \(b > 0\) by

\(\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\).

This function returns its log,

\(\log \mbox{B}(a, b) = \log \Gamma(a) + \log \Gamma(b) - \log \Gamma(a+b)\).

See stan::math::lgamma() for the double-based and stan::math for the variable-based log Gamma function. This function is numerically more stable than naive evaluation via lgamma.

\[ \mbox{lbeta}(\alpha, \beta) = \begin{cases} \ln\int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lbeta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \Psi(\alpha)-\Psi(\alpha+\beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lbeta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \Psi(\beta)-\Psi(\alpha+\beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Parameters

a	First value
b	Second value

Returns

Log of the beta function applied to the two values.

Definition at line 62 of file lbeta.hpp.