The MCMC Procedure
Standard Distributions
The section Univariate Distributions (Table 7 through Table 36) lists all univariate distributions that PROC MCMC recognizes. The section Multivariate Distributions (Table 37 through Table 41) lists all multivariate distributions that PROC MCMC recognizes. With the exception of the multinomial distribution, all these distributions can be used in the MODEL, PRIOR, and HYPERPRIOR statements. The multinomial distribution is supported only in the MODEL statement. The RANDOM statement supports a limited number of distributions; see Table 4 for the complete list.
See the section Using Density Functions in the Programming Statements for information about how to use distributions in the programming statements. To specify an arbitrary distribution, you can use the GENERAL and DGENERAL functions. See the section Specifying a New Distribution for more details. See the section Truncation and Censoring for tips about how to work with truncated distributions and censoring data.
Univariate Distributions
Table 7: Beta Distribution
Table 8: Binary Distribution
| PROC specification | binary(p) |
| Density |  |
| Parameter restriction |  |
| Range |  |
| Mean |  |
| Variance |  |
| Mode |  |
| Random number | Generate  . If  ,  ; else,  |
Table 9: Binomial Distribution
| PROC specification | binomial(n, p) |
| Density |  |
| Parameter restriction |  |
| Range |  |
| Mean |  |
| Variance |  |
| Mode |  |
Table 10: Cauchy Distribution
| PROC specification | cauchy(a, b) |
| Density |  |
| Parameter restriction |  |
| Range |  |
| Mean | Does not exist. |
| Variance | Does not exist. |
| Mode | a |
| Random number | Generate  ; let  . Repeat the procedure until  .  is a draw from the standard Cauchy, and  (Ripley 1987). |
Table 11: 
Distribution
| PROC specification | chisq( ) |
| Density |  |
| Parameter restriction |  |
| Range |  if  ;  otherwise. |
| Mean |  |
| Variance |  |
| Mode |  if  ; does not exist otherwise. |
| Random number | is a special case of the gamma distribution:  is a draw from the distribution. |
Table 12: Exponential Distribution
| PROC specification | expchisq() |
| Density |  |
| Parameter restriction | |
| Range | |
| Mode |  |
| Random number | Generate  , and  is a draw from the exponential distribution. |
| Relationship to the distribution |
 |
Table 13: Exponential Exponential Distribution
| PROC specification | expexpon(scale = b ) |
expexpon(iscale =  ) |
| Density |  |
 |
| Parameter restriction |  |
 |
| Range |  |
Same |
| Mode |  |
 |
| Random number | Generate  , and is a draw from the exponential exponential distribution. Note that an exponential exponential distribution is not the same as the double exponential distribution. |
|
| Relationship to the exponential distribution |  |
|
Table 14: Exponential Gamma Distribution
| PROC specification | expgamma(a, scale = b ) |
expgamma(a, iscale = ) |
| Density |  |
 |
| Parameter restriction |  |
 |
| Range | |
Same |
| Mode |  |
 |
| Random number | Generate  , and is a draw from the exponential gamma distribution. |
|
Relationship to the  distribution |
 |
|
Table 15: Exponential Inverse Distribution
| PROC specification | expichisq() |
| Density |  |
| Parameter restriction | |
| Range | |
| Mode |  |
| Random number | Generate  , and is a draw from the exponential inverse distribution. |
Relationship to the  distribution |
 |
Table 16: Exponential Inverse Gamma Distribution
| PROC specification | expigamma(a, scale = b ) |
expigamma(a, iscale = ) |
| Density |  |
 |
| Parameter restriction | |
|
| Range | |
Same |
| Mode |  |
 |
| Random number | Generate  , and is a draw from the exponential inverse gamma distribution. |
|
Relationship to the  distribution |
 |
|
Table 17: Exponential Scaled Inverse Distribution
| PROC specification | expsichisq(, s) |
| Density |  |
| Parameter restriction |  |
| Range | |
| Mode |  |
| Random number | Generate  , and is a draw from the exponential scaled inverse distribution. |
Relationship to the  distribution |
 |
Table 18: Exponential Distribution
| PROC specification | expon(scale = b ) |
expon(iscale = ) |
| Density |  |
 |
| Parameter restriction | |
|
| Range | |
Same |
| Mean | b |  |
| Variance |  |
 |
| Mode | 0 | 0 |
| Random number | The exponential distribution is a special case of the gamma distribution:  is a draw from the exponential distribution. |
|
Table 19: Gamma Distribution
| PROC specification | gamma(a, scale = b ) |
gamma(a, iscale = ) |
| Density |  |
 |
| Parameter restriction | |
|
| Range |  if  otherwise. |
Same |
| Mean | ab |  |
| Variance |  |
 |
| Mode |  if  |
 if |
| Random number | See (McGrath and Irving 1973). | |
Table 20: Geometric Distribution
| PROC specification | geo(p) |
| Density * |  |
| Parameter restriction |  |
| Range | 
|
| Mean |  |
| Variance |  |
| Mode | 0 |
| Random number | Based on samples obtained from a Bernoulli distribution with probability p until the first success. |
*The random variable  is the total number of failures in an experiment before the first success. This density function is not to be confused with another popular formulation,  , which counts the total number of trials until the first success. |
|
Table 21: Inverse Distribution
| PROC specification | ichisq() |
| Density |  |
| Parameter restriction | |
| Range |  |
| Mean |  if  |
| Variance |  if  |
| Mode |  |
| Random number | Inverse is a special case of the inverse gamma distribution:  is a draw from the inverse distribution. |
Table 22: Inverse Gamma Distribution
| PROC specification | igamma(a, scale = b ) |
igamma(a, iscale = ) |
| Density |  |
 |
| Parameter restriction | |
|
| Range | |
Same |
| Mean |  if  |
 if |
| Variance |  |
 |
| Mode |  |
 |
| Random number | Generate , and  is a draw from the  distribution. |
|
| Relationship to the gamma distribution |  |
|
Table 23: Laplace (Double Exponential) Distribution
| PROC specification | laplace(a, scale = b) |
laplace(a, iscale = ) |
| Density |  |
 |
| Parameter restriction | |
 |
| Range | |
Same |
| Mean | a | a |
| Variance |  |
 |
| Mode | a | a |
| Random number | Inverse CDF.  Generate  . If  else  . is a draw from the Laplace distribution. |
|
Table 24: Logistic Distribution
| PROC specification | logistic(a, b) |
| Density |  |
| Parameter restriction | |
| Range | |
| Mean | a |
| Variance |  |
| Mode | a |
| Random number | Inverse CDF method with  . Generate  , and  is a draw from the logistic distribution. |
Table 25: Lognormal Distribution
| PROC specification | lognormal( , sd = s) |
lognormal(, var = v) |
lognormal(, prec =  ) |
| Density |  |
 |
 |
| Parameter restriction |  |
 |
 |
| Range | |
Same | Same |
| Mean |  |
 |
 |
| Variance |  |
 |
 |
| Mode |  |
 |
 |
| Random number | Generate  , and  is a draw from the lognormal distribution. |
||
Table 26: Negative Binomial Distribution
| PROC specification | negbin(n, p) |
| Density |  |
| Parameter restriction |  |
| Range |
|
| Mean | round |
| Variance |  |
| Mode |  |
| Random number | Generate  , and  (Fishman 1996). |
Table 27: Normal Distribution
| PROC specification | normal(, sd = s) |
normal(, var = v) |
normal(, prec = ) |
| Density |  |
 |
 |
| Parameter restriction | |
|
|
| Range | |
Same | Same |
| Mean | |
Same | Same |
| Variance |  |
v |  |
| Mode | |
Same | Same |
Table 28: NormalCAR Distribution
| PROC specification | normalcar(neighbors=, num=, sd= s) |
normalcar(neighbors=, num=, var= v) |
normalcar(neighbors=, num=, prec= ) |
| Density |  |
 |
 |
| Notation | i is the area or site index,  is the number of neighbors to i, and  is the set of neighbors to i. |
||
| Parameter restriction | |
|
|
| Range | |
Same | Same |
| Mean | average of neighbors | Same | Same |
| Variance | |
v | |
| Mode | average of neighbors | Same | Same |
Table 29: Pareto Distribution
| PROC specification | pareto(a, b) |
| Density |  |
| Parameter restriction | |
| Range |  |
| Mean |  if  |
| Variance |  if  |
| Mode | b |
| Random number | Inverse CDF method with  . Generate , and  is a draw from the Pareto distribution. |
| Useful transformation |  is Beta(a, 1)I{ }. |
Table 30: Poisson Distribution
| PROC specification | poisson( ) |
| Density |  |
| Parameter restriction |  |
| Range | 
|
| Mean | |
| Variance | , if  |
| Mode | round |
Table 31: Scaled Inverse Distribution
| PROC specification | sichisq( ) |
| Density |  |
| Parameter restriction | |
| Range | |
| Mean |  if |
| Variance |  if |
| Mode |  |
| Random number | Scaled inverse is a special case of the inverse gamma distribution:  is a draw from the scaled inverse distribution. |
Table 32: t Distribution
| PROC specification | t(, sd = s, ) |
t(, var = v, ) |
t(, prec = , ) |
| Density |  |
 |
 |
| Parm restriction | , |
, |
, |
| Range | |
Same | Same |
| Mean |  if  |
Same | Same |
| Variance |  if  |
 if |
 if |
| Mode | |
Same | Same |
| Random number |  is a draw from the t distribution. |
||
Table 33: Table (Categorical) Distribution
| PROC specification | table( ), where  , for  |
| Density |  |
| Parameter restriction |  with all  |
| Range |  |
| Mode | i such that  |
| Random number | Inverse CDF method with  . |
Table 34: Uniform Distribution
Table 35: Wald Distribution
| PROC specification | wald(, ) |
| Density |  |
| Parameter restriction |  |
| Range | |
| Mean | |
| Variance |  |
| Mode |  |
| Random number | Generate  . Let  and  . Perform a Bernoulli trial,  . If  , choose  ; otherwise, choose  (Michael, Schucany, and Haas 1976). |
Table 36: Weibull Distribution
| PROC specification | weibull(, c,  ) |
| Density |  |
| Parameter restriction |  |
| Range |  if  otherwise |
| Mean |  |
| Variance |  |
| Mode |  if  |
| Random number | Inverse CDF method with  . Generate , and  is a draw from the Weibull distribution. |
Multivariate Distributions
Table 37: Dirichlet Distribution
| PROC specification |  dirich( ), where  , for  |
| Density |  , where  |
| Parameter restriction | 
|
| Range |  ,  |
| Mean |  |
| Mode |  |
Table 38: Inverse Wishart Distribution
| PROC specification | iwishart(,  ), both  and are  matrices |
| Density |  |
| Parameter restriction | must be symmetric and positive definite;  |
| Range | is symmetric and positive definite |
| Mean |  |
| Mode |  |
Table 39: Multivariate Normal Distribution
| PROC specification | mvn( ,  ), where  , for  , and is a variance matrix |
| Density |  |
| Parameter restriction | must be symmetric and positive definite |
| Range |  |
| Mean |
|
| Mode |
|
Table 40: Autoregressive Multivariate Normal Distribution
| PROC specification |  MVNAR(, sd=, ) |
MVNAR(, var= ,) |
MVNAR(, prec= , ) |
| Density |  where  |
||
| Parameter restriction |  and  |
||
| Range | |
||
| Mean | |
||
| Mode | |
||
| Special Case | When  , the distribution simplifies to mvn(,  ), where  denotes the identity matrix |
||
Table 41: Multinomial Distribution
| PROC specification | multinom(), where  and  , for |
| Density |  , where 
|
| Parameter restriction | with all |
| Range |  , nonnegative integers |
| Mean | 
|