HPQLIM Procedure

The PRIOR statement specifies the prior distribution of the model parameters. You must specify one parameter or a list of parameters, a tilde , 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) and 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 3 are considered.

Table 3: Default Values for Prior Distributions

PRIOR Distribution					Parameters Default Choice
NORMAL	MEAN=0	VAR=1E6
IGAMMA	SHAPE=2.000001	SCALE=1
GAMMA	SHAPE=1	SCALE=1
UNIFORM
BETA	SHAPE1=1	SHAPE2=1
T	LOCATION=0	DF=3

For density specification, see the section Standard Distributions.

Standard Distributions

Table 4 through Table 9 show all the distribution density functions that PROC HPQLIM recognizes. You specify these distribution densities in the PRIOR statement.

Table 4: Beta Distribution

PRIOR statement	BETA(SHAPE1=a, SHAPE2=b, MIN=m, MAX=M)
	Note: Commonly and .
Density
Parameter restriction	, ,
Range
Mean
Variance
Mode
Defaults	SHAPE1=SHAPE2=1, ,

Table 5: Gamma Distribution

PRIOR statement	GAMMA(SHAPE=a, SCALE=b)
Density
Parameter restriction
Range
Mean
Variance
Mode
Defaults	SHAPE=SCALE=1

Table 6: Inverse Gamma Distribution

PRIOR statement	IGAMMA(SHAPE=a, SCALE=b)
Density
Parameter restriction
Range
Mean
Variance
Mode
Defaults	SHAPE=2.000001, SCALE=1

Table 7: Normal Distribution

PRIOR statement	NORMAL(MEAN=, VAR=)
Density
Parameter restriction
Range
Mean
Variance
Mode
Defaults	MEAN=0, VAR=1000000

Table 8: t Distribution

PRIOR statement	T(LOCATION=, DF=)
Density
Parameter restriction
Range
Mean
Variance
Mode
Defaults	LOCATION=0, DF=3

Table 9: Uniform Distribution

PRIOR statement	UNIFORM(MIN=m, MAX=M)
Density
Parameter restriction
Range
Mean
Variance
Mode	Not unique
Defaults	MIN, MAX