# 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, $\begin{equation*} \Psi(u) \ = \ \frac{d}{d u} \log \Gamma(u) \ = \ \frac{1}{\Gamma(u)} \ \frac{d}{d u} \Gamma(u). \end{equation*}$