The MCMC Procedure

Using Density Functions in the Programming Statements

Density Functions in PROC MCMC
Standard Distributions, the LOGPDF Functions, and the LPDFdist Functions

Density Functions in PROC MCMC

PROC MCMC has a number of internally defined log-density functions for univariate and multivariate distributions. These functions have the basic form of LPDFdist(x, parm-list), where dist is the name of the distribution (see Table 55.40 for univariate distributions and Table 55.41 for multivariate distributions). The argument x is the random variable, and parm-list is the list of parameters.

In addition, the univariate functions allow for optional boundary arguments, such as LPDFdist(x, parm-list, <lower>, <upper>), where lower and upper are optional but positional boundary arguments. With the exception of the Bernoulli and uniform distribution, you can specify limits on all univariate distributions.

To set a lower bound on the normal density:

lpdfnorm(x, 0, 1, -2);

To set just an upper bound, specify a missing value for the lower bound argument:

lpdfnorm(x, 0, 1, ., 2);

Leaving both limits out gives you the unbounded density. You can also specify both bounds:

lpdfnorm(x, 0, 1);
lpdfnorm(x, 0, 1, -3, 4);

See Table 55.40 for the function names of univariate distributions and Table 55.41 for multivariate distributions.

Table 55.40: Logarithm of Univariate Density Functions in PROC MCMC

Distribution Name	Function Call
Beta	`lpdfbeta` (x, a, b,<lower>, <upper>) ;
Binary	`lpdfbern` (x, p) ;
Binomial	`lpdfbin` (x, n,p, <lower>, <upper>) ;
Cauchy	`lpdfcau` (x, loc, scale, <lower>, <upper>) ;
$\chi ^2$	`lpdfchisq` (x, df,<lower>, <upper>) ;
Exponential $\chi ^2$	`lpdfechisq` (x, df, <lower>, <upper>) ;
Exponential gamma	`lpdfegamma` (x, sp,scale, <lower>, <upper>) ;
Exponential exponential	`lpdfeexpon` (x, scale,<lower>, <upper>) ;
Exponential inverse $\chi ^2$	`lpdfeichisq` (x, df, <lower>, <upper>) ;
Exponential inverse-gamma	`lpdfeigamma` (x, sp, scale, <lower>, <upper>) ;
Exponential scaled inverse $\chi ^2$	`lpdfesichisq` (x, df, scale, <lower>, <upper>) ;
Exponential	`lpdfexpon` (x, scale, <lower>, <upper>) ;
Gamma	`lpdfgamma` (x, sp, scale, <lower>, <upper>) ;
Geometric	`lpdfgeo` (x, p, <lower>, <upper>) ;
Inverse $\chi ^2$	`lpdfichisq` (x, df, <lower>, <upper>) ;
Inverse-gamma	`lpdfigamma` (x, sp, scale, <lower>, <upper>) ;
Laplace	`lpdfdexp` (x, loc, scale, <lower>, <upper>) ;
Logistic	`lpdflogis` (x, loc, scale, <lower>, <upper>) ;
Lognormal	`lpdflnorm` (x, loc, sd, <lower>, <upper>) ;
Negative binomial	`lpdfnegbin` (x, n, p, <lower>, <upper>) ;
Normal	`lpdfnorm` (x, mu, sd, <lower>, <upper>) ;
Pareto	`lpdfpareto` (x, sp, scale, <lower>, <upper>) ;
Poisson	`lpdfpoi` (x, mean, <lower>, <upper>) ;
Scaled inverse $\chi ^2$	`lpdfsichisq` (x, df, scale, <lower>, <upper>) ;
t	`lpdft` (x, mu, sd, df, <lower>,<upper>) ;
Uniform	`lpdfunif` (x, a, b) ;
Wald	`lpdfwald` (x, mean, scale, <lower>, <upper>) ;
Weibull	`lpdfwei` (x, loc, sp, scale, <lower>, <upper>) ;

In the multivariate log-density functions, arrays must be used in place for the random variable and parameters in the model.

Table 55.41: Logarithm of Multivariate Density Functions in PROC MCMC

Distribution Name	Function Call
Dirichlet	`lpdfdirch` (x_array, alpha_array) ;
Inverse Wishart	`lpdfiwish` (x_array, df, S_array) ;
Multivariate normal	`lpdfmvn` (x_array, mu_array, cov_array) ;
Multinomial	`lpdfmnom` (x_array, p_array) ;

Standard Distributions, the LOGPDF Functions, and the LPDFdist Functions

Standard distributions listed in the section Standard Distributions are names only, and they can be used only in the MODEL, PRIOR, and HYPERPRIOR statements to specify either a prior distribution or a conditional distribution of the data given parameters. They do not return any values, and you cannot use them in the programming statements.

The LOGPDF functions are DATA step functions that compute the logarithm of various probability density (mass) functions. For example,

logpdf ("beta", x, 2, 15) ;

returns the log of a beta density with parameters a = 2 and b = 15, evaluated at x. All the LOGPDF functions are supported in PROC MCMC.

The LPDFdist functions are unique to PROC MCMC. They compute the logarithm of various probability density (mass) functions. The functions are the same as the LOGPDF functions when it comes to calculating the log density. For example,

lpdfbeta (x, 2,15) ;

returns the same value as

logpdf ("beta", x, 2, 15) ;

The LPDFdist functions cover a greater class of probability density functions, and the univariate distribution functions take the optional but positional boundary arguments. There are no corresponding LCDFdist or LSDFdist functions in PROC MCMC. To work with the cumulative probability function or the survival functions, you need to use the LOGCDF and the LOGSDF DATA step functions.