lbeta() [6/10]

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t< T1, T2 > stan::math::lbeta	(	const T1	a,
		const T2	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.

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

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

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

Template Parameters

T1	type of first value
T2	type of second value

Parameters

a	First value
b	Second value

Returns

Log of the beta function applied to the two values.

Definition at line 65 of file lbeta.hpp.