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 $$\text{B}(a,b)={\int }_0^1{u}^{a-1}(1-u{)}^{b-1}du=\frac{\mathrm{\Gamma }(a)\mathrm{\Gamma }(b)}{\mathrm{\Gamma }(a+b)},$$ where $\mathrm{\Gamma }(x)$ is the Gamma function.

Incomplete beta

The incomplete beta function, $\text{B}(x;a,b)$, is defined for $x\in [0,1]$ and $a,b\ge 0$ such that $a+b\ne 0$ by $$\text{B}(x;a,b)={\int }_0^x{u}^{a-1}(1-u{)}^{b-1}du,$$ 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, $$I_x(a,b)=\frac{\text{B}(x;a,b)}{B(a,b)}.$$

Gamma

The gamma function, $\mathrm{\Gamma }(x)$, is the generalization of the factorial function to continuous variables, defined so that for positive integers $n$, $$\mathrm{\Gamma }(n+1)=n!$$ Generalizing to all positive numbers and non-integer negative numbers, $$\mathrm{\Gamma }(x)={\int }_0^{\mathrm{\infty }}{u}^{x-1}\exp (-u)du.$$

Digamma

The digamma function $\mathrm{\Psi }$ is the derivative of the $\log \mathrm{\Gamma }$ function, $$\mathrm{\Psi }(u)=\frac{d}{du}\log \mathrm{\Gamma }(u)=\frac{1}{\mathrm{\Gamma }(u)}\frac{d}{du}\mathrm{\Gamma }(u).$$