Mathematical Functions

This appendix provides the definition of several mathematical functions used throughout the manual.

Beta

The beta function, \(\text{B}(a, b)\), computes the normalizing constant for the beta distribution, and is defined for \(a > 0\) and \(b > 0\) by \[\begin{equation*} \text{B}(a,b) \ = \ \int_0^1 u^{a - 1} (1 - u)^{b - 1} \, du \ = \ \frac{\Gamma(a) \, \Gamma(b)}{\Gamma(a+b)} \, , \end{equation*}\] where \(\Gamma(x)\) is the Gamma function.

Incomplete beta

The incomplete beta function, \(\text{B}(x; a, b)\), is defined for \(x \in [0, 1]\) and \(a, b \geq 0\) such that \(a + b \neq 0\) by \[\begin{equation*} \text{B}(x; \, a, b) \ = \ \int_0^x u^{a - 1} \, (1 - u)^{b - 1} \, du, \end{equation*}\] where \(\text{B}(a, b)\) is the beta function defined in appendix. If \(x = 1\), the incomplete beta function reduces to the beta function, \(\text{B}(1; a, b) = \text{B}(a, b)\).

The regularized incomplete beta function divides the incomplete beta function by the beta function, \[\begin{equation*} I_x(a, b) \ = \ \frac{\text{B}(x; \, a, b)}{B(a, b)} \, . \end{equation*}\]

Gamma

The gamma function, \(\Gamma(x)\), is the generalization of the factorial function to continuous variables, defined so that for positive integers \(n\), \[\begin{equation*} \Gamma(n+1) = n! \end{equation*}\] Generalizing to all positive numbers and non-integer negative numbers, \[\begin{equation*} \Gamma(x) = \int_0^{\infty} u^{x - 1} \exp(-u) \, du. \end{equation*}\]

Digamma

The digamma function \(\Psi\) is the derivative of the \(\log \Gamma\) function, \[\begin{equation*} \Psi(u) \ = \ \frac{d}{d u} \log \Gamma(u) \ = \ \frac{1}{\Gamma(u)} \ \frac{d}{d u} \Gamma(u). \end{equation*}\]