The HPQLIM Procedure

Prior Distributions

Subsections:

Standard Distributions

The PRIOR statement specifies the prior distribution of the model parameters. You must specify one parameter or a list of parameters, a tilde ${\scriptstyle \sim }$ , and then a distribution with its parameters. You can specify multiple PRIOR statements to define independent priors. Parameters that are associated with a regressor variable are referred to by the name of the corresponding regressor variable.

You can specify the special keyword _REGRESSORS to consider all the regressors of a model. If multiple PRIOR statements affect the same parameter, the last PRIOR statement prevails. For example, in a regression with two regressors (X1, X2), the following statements imply that the prior on X1 is NORMAL(MEAN=0, VAR=1), the prior on X2 is GAMMA(SHAPE=3, SCALE=4).

...
prior _Regressors ~ uniform(min=0, max=1);
prior X1 X2 ~ gamma(shape=3, scale=4);
prior X1 ~ normal(mean=0, var=1);
...

If a parameter is not associated with a PRIOR statement or if some of the prior hyperparameters are missing, then the default choices in Table Table 8.3 are considered.

Table 8.3: Default Values for Prior Distributions

PRIOR Distribution	$\Variable{Hyperparameter}_1$	$\Variable{Hyperparameter}_2$	$\Variable{Min}$	$\Variable{Max}$	`Parameters Default Choice`
NORMAL	`MEAN=0`	`VAR=1E6`	$-\infty$	$\infty$	$\Variable{Regression-Location-Threshold}$
IGAMMA	`SHAPE=2.000001`	`SCALE=1`	$>0$	$\infty$	$\Variable{Scale}$
GAMMA	`SHAPE=1`	`SCALE=1`	$\Variable{0}$	$\infty$
UNIFORM			$-\infty$	$\infty$
BETA	`SHAPE`1=1	`SHAPE`2=1	$-\infty$	$\infty$
T	`LOCATION`=0	`DF`=3	$-\infty$	$\infty$

For density specification, see the section Standard Distributions.

Standard Distributions

Table 8.4 through Table 8.9 show all the distribution density functions that PROC HPQLIM recognizes. You specify these distribution densities in the PRIOR statement.

Table 8.4: Beta Distribution

PRIOR statement	BETA(SHAPE1=a, SHAPE2=b, MIN=m, MAX=M)
	Note: Commonly $m=0$ and $M=1$ .
Density	$\frac{(\theta -m)^{a-1} (M-\theta )^{b-1}}{B(a,b)(M-m)^{a+b-1}}$
Parameter restriction	$a>0$ , $b>0$ , $-\infty <m<M<\infty$
Range	$\left\{ \begin{array}{ll} \left[ m, M \right] & \mbox{when } a = 1, b = 1 \\ \left[ m, M \right) & \mbox{when } a = 1, b \neq 1 \\ \left( m, M \right] & \mbox{when } a \neq 1, b = 1 \\ \left( m, M \right) & \mbox{otherwise} \end{array} \right.$
Mean	$\frac{a}{a+b}\times (M-m)+m$
Variance	$\frac{ab}{(a+b)^2(a+b+1)}\times (M-m)^2$
Mode	$\left\{ \begin{array}{ll} \frac{a-1}{a+b-2}\times M+\frac{b-1}{a+b-2}\times m & a > 1, b > 1 \\ m \mbox{ and } M & a < 1, b < 1 \\ m & \left\{ \begin{array}{l} a < 1, b \geq 1 \\ a = 1, b > 1 \\ \end{array} \right. \\ M & \left\{ \begin{array}{l} a \geq 1, b < 1 \\ a > 1, b = 1 \\ \end{array} \right. \\ \mbox{not unique} & a = b = 1 \end{array} \right.$
Defaults	SHAPE1=SHAPE2=1, $\Variable{MIN}\rightarrow -\infty$ , $\Variable{MAX}\rightarrow \infty$

Table 8.5: Gamma Distribution

PRIOR statement	GAMMA(SHAPE=a, SCALE=b)
Density	$\frac{1}{b^ a\Gamma (a)} \theta ^{a-1} e^{-\theta /b}$
Parameter restriction	$a > 0, b > 0$
Range	$[0,\infty )$
Mean	$ab$
Variance	$ab^2$
Mode	$(a-1)b$
Defaults	SHAPE=SCALE=1

Table 8.6: Inverse Gamma Distribution

PRIOR statement	IGAMMA(SHAPE=a, SCALE=b)
Density	$\frac{b^ a}{\Gamma (a)} \theta ^{-(a+1)}e^{-b/\theta }$
Parameter restriction	$a > 0, b > 0$
Range	$0<\theta <\infty$
Mean	$\frac{b}{a-1},\qquad a > 1$
Variance	$\frac{b^2}{(a-1)^2(a-2)},\qquad a>2$
Mode	$\frac{b}{a+1}$
Defaults	SHAPE=2.000001, SCALE=1

Table 8.7: Normal Distribution

PRIOR statement	NORMAL(MEAN= $\mu$ , VAR= $\sigma ^2$ )
Density	$\frac{1}{\sigma \sqrt {2\pi }} \exp \left( - \frac{(\theta - \mu )^2}{2\sigma ^2}\right)$
Parameter restriction	$\sigma ^2 > 0$
Range	$-\infty <\theta <\infty$
Mean	$\mu$
Variance	$\sigma ^2$
Mode	$\mu$
Defaults	MEAN=0, VAR=1000000

Table 8.8: t Distribution

PRIOR statement	T(LOCATION= $\mu$ , DF= $\nu$ )
Density	$\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)\sqrt {\pi \nu }}\left[1+\frac{(\theta -\mu )^2}{\nu }\right]^{-\frac{\nu +1}{2}}$
Parameter restriction	$\nu > 0$
Range	$-\infty <\theta <\infty$
Mean	$\mu , \text { for }\nu >1$
Variance	$\frac{\nu }{\nu -2}, \text { for }\nu >2$
Mode	$\mu$
Defaults	LOCATION=0, DF=3

Table 8.9: Uniform Distribution

PRIOR statement	UNIFORM(MIN=m, MAX=M)
Density	$\frac{1}{M-m}$
Parameter restriction	$-\infty <m<M<\infty$
Range	$\theta \in [m, M]$
Mean	$\frac{m+M}{2}$
Variance	$\frac{(M-m)^2}{12}$
Mode	Not unique
Defaults	MIN $\rightarrow -\infty$ , MAX $\rightarrow \infty$