CQLIM Procedure
Prior Distributions
The PRIOR statement is used to specify the prior distribution of the model parameters. You must specify a list of parameters, a tilde (~), and then a distribution with its parameters. You can specify multiple PRIOR statements to define independent prior distributions. Parameters that are associated with a regressor variable are referred to by the name of the corresponding regressor variable. For information about how parameters are named so that you can refer to them when specifying prior distributions, see the section Naming of Parameters. Alternatively, the SHOWNAMES suboption of the PRIORSUMMARY option displays the names that are used in all parameters in the prior summary table.
If a parameter is not associated with a PRIOR statement, then the default choices shown in Table 4 are used.
Table 4: Default Values for Prior Distributions
| Parameter Type | Default Prior Name | Default Prior Specification |
|---|---|---|
| Regression, location | Normal | NORMAL(MEAN=0, VAR=1E6) |
| Scale, SD | Truncated normal | NORMAL(MEAN=0, VAR=1E6, LOWER=0, UPPER=INF) |
For more information about each prior density, see the section Standard Prior Distributions in Chapter 2, Introduction to Bayesian Analysis Procedures. For information about truncated densities, see the section Truncated Prior Distributions. For information about why the default prior on the standard deviation parameter is a truncated normal distribution, see the section Prior Distributions for Scale Parameters.
Prior Distributions for Scale Parameters
A common choice for a prior distribution on variance parameters and other scale parameters is the inverse gamma distribution. This prior can often be problematic in practice, and as a consequence it can even cause computational problems. This is partially because the density puts essentially no probability mass near zero. For more information, see Gelman (2006). The main advantage of the inverse gamma prior is that it is the conditionally conjugate prior for a variance parameter in a normal model; this makes certain sampling algorithms easier to use. For more information, see the section Conjugate Priors in Chapter 2, Introduction to Bayesian Analysis Procedures. However, the posterior sampler in PROC CQLIM does not take advantage of this conjugacy, so there is no computational benefit to using the inverse gamma prior.
Instead, PROC CQLIM uses the half-normal distribution for the default prior on standard deviation parameters—that is, a mean zero normal distribution truncated from below at zero. This density allows for the standard deviation parameter to be estimated arbitrarily close to zero and otherwise removes the problems associated with the inverse gamma prior. In general, if you want to specify prior information on the standard deviation parameter or some other scale parameter, it is recommended that you encode that information into a truncated normal distribution or perhaps a truncated t distribution.
If you want to specify the classic inverse gamma prior for a variance parameter, note that PROC CQLIM is parameterized in terms of the standard deviation parameter. So instead, the square root inverse gamma prior should be used on _Sigma, the standard deviation parameter in most models in PROC CQLIM.
Truncated Prior Distributions
All prior distributions except the uniform distribution can be turned into truncated prior distributions by using the LOWER and UPPER parameters. By default, LOWER is set to the maximum of the parameter’s lower bound and the specified prior distribution’s lower bound, and UPPER is set to the minimum of the parameter’s upper bound and the specified prior distribution’s upper bound. For example, for _Sigma, the standard deviation parameter in a regression model, the following statement specifies a truncated normal prior on _Sigma with LOWER=0 and UPPER=INFINITY:
prior _Sigma ~ normal(mean = 0, var = 1);
The density for of truncated prior can be computed from the corresponding untruncated density given in the section Standard Prior Distributions in Chapter 2, Introduction to Bayesian Analysis Procedures, as follows. Suppose 
is the prior density for the parameter 
. Then let 
denote the corresponding CDF, defined by the integral
where 
is taken to be 0 outside its range (for each prior density’s range, see the section Standard Prior Distributions in Chapter 2, Introduction to Bayesian Analysis Procedures). Then the truncated density is constructed from the untruncated density by using the formula
where 
are the lower and upper truncation bounds, respectively. If the lower truncation bound is the lower bound of 
’s range, then 
, and similarly if the upper truncation bound is the upper bound of ’s range, then 
.