Introduction to Bayesian Analysis Procedures
Standard Prior Distributions
Table 1 through Table 10 show the density functions for all the distributions that you can use as priors in SAS Econometrics procedures. Some procedures support only a subset of these distributions. You can typically specify these distributions as priors for model parameters in a PRIOR statement.
Table 1: Beta Distribution
| Density |  |
| Parameter restriction |  , 
|
| Range |  |
| Mean |  |
| Variance |  |
| Mode |  |
Table 2: Cauchy Distribution
| Density |  |
| Parameter restriction | 
|
| Range | 
|
| Mean | Not defined |
| Variance | Not defined |
| Mode |  |
Table 3: Gamma Distribution
| Density |  |
| Parameter restriction | 
|
| Range |  |
| Mean |  |
| Variance |  |
| Mode |  |
Table 4: Inverse Gamma Distribution
| Density |  |
| Parameter restriction | 
|
| Range | 
|
| Mean |  |
| Variance |  |
| Mode |  |
Table 5: Lognormal Distribution
| Density |  |
| Parameter restriction | 
|
| Range | 
|
| Mean |  |
| Variance |  |
| Mode |  |
Table 6: Normal Distribution
| Density |  |
| Parameter restriction |
|
| Range | 
|
| Mean | |
| Variance |  |
| Mode | |
Table 7: Square Root Gamma Distribution
| Density |  |
| Parameter restriction |
|
| Range | |
| Mean |  |
| Variance |  |
| Mode |  |
| This is also known as the Nakagami distribution. For more information, see Stacy (1962). | |
Table 8: Square Root Inverse Gamma Distribution
| Density |  |
| Parameter restriction |
|
| Range |
|
| Mean |  |
| Variance |  |
| Mode |  |
| This is also known as the inverse Nakagami distribution. For more information, see Stacy (1962). | |
Table 9: t Distribution
| Density |  |
| Parameter restriction | , 
|
| Range | 
|
| Mean |  |
| Variance |  |
| Mode | |
Table 10: Uniform Distribution
| Density |  |
| Parameter restriction |  |
| Range |  |
| Mean |  |
| Variance |  |
| Mode | Not unique |
Some procedures allow for improper uniform priors, where  or  . For more information, see the section Improper Priors. | |