PRIOR <distribution> </ options>;
The PRIOR statement enables you to carry out a samplingbased Bayesian analysis in PROC MIXED. It currently operates only with variance component models. Other TYPE= structures are not supported. The analysis produces a SAS data set containing a pseudorandom sample from the joint posterior density of the variance components and other parameters in the mixed model.
The posterior analysis is performed after all other PROC MIXED computations. It begins with the "Posterior Sampling Information" table, which provides basic information about the posterior sampling analysis, including the prior densities, sampling algorithm, sample size, and random number seed. The ODS name of this table is Posterior.
By default, PROC MIXED uses an independence chain algorithm in order to generate the posterior sample (Tierney 1994). This algorithm works by generating a pseudorandom proposal from a convenient base distribution, chosen to be as close as possible to the posterior. The proposal is then retained in the sample with probability proportional to the ratio of weights constructed by taking the ratio of the true posterior to the base density. If a proposal is not accepted, then a duplicate of the previous observation is added to the chain.
In selecting the base distribution, PROC MIXED makes use of the fact that the fixedeffects parameters can be analytically integrated out of the joint posterior, leaving the marginal posterior density of the variance components. In order to better approximate the marginal posterior density of the variance components, PROC MIXED transforms them by using the MIVQUE(0) equations. You can display the selected transformation with the PTRANS option or specify your own with the TDATA= option. The density of the transformed parameters is then approximated by a product of inverted gamma densities (see Gelfand et al. 1990).
To determine the parameters for the inverted gamma densities, PROC MIXED evaluates the logarithm of the posterior density over a grid of points in each of the transformed parameters, and you can display the results of this search with the PSEARCH option. PROC MIXED then performs a linear regression of these values on the logarithm of the inverted gamma density. The resulting base densities are displayed in the "Base Densities" table; the ODS name of this table is Base. You can input different base densities with the BDATA= option.
At the end of the sampling, the "Acceptance Rates" table displays the acceptance rate computed as the number of accepted samples divided by the total number of samples generated. The ODS name of the "Acceptance Rates" table is AccRates.
The OUT= option specifies the output data set containing the posterior sample. PROC MIXED automatically includes all variance component parameters in this data set (labeled COVP1–COVPn), the Type 3 F statistics constructed as in Ghosh (1992) discussing Schervish (1992) (labeled T3Fn), the log values of the posterior (labeled LOGF), the log of the base sampling density (labeled LOGG), and the log of their ratio (labeled LOGRATIO). If you specify the SOLUTION option in the MODEL statement, the data set also contains a random sample from the posterior density of the fixedeffects parameters (labeled BETAn); and if you specify the SOLUTION option in the RANDOM statement, the table contains a random sample from the posterior density of the randomeffects parameters (labeled GAMn). PROC MIXED also generates additional variables corresponding to any CONTRAST , ESTIMATE , or LSMEANS statement that you specify.
Subsequently, you can use SAS/INSIGHT or the UNIVARIATE, CAPABILITY, or KDE procedure to analyze the posterior sample.
The prior density of the variance components is, by default, a noninformative version of Jeffreys’ prior (Box and Tiao 1973). You can also specify informative priors with the DATA= option or a flat (equal to 1) prior for the variance components. The prior density of the fixedeffects parameters is assumed to be flat (equal to 1), and the resulting posterior is conditionally multivariate normal (conditioning on the variance component parameters) with mean and variance .
Table 77.14 summarizes the options available in the PRIOR statement.
Table 77.14: PRIOR Statement Options
Option 
Description 

Inputs the prior densities of the variance components 

Specifies a noninformative reference version of Jeffreys’ prior 

Specifies a prior density equal to 1 everywhere 

Specifies the algorithm used for generating the posterior sample 

Inputs the base densities used by the sampling algorithm 

Specifies a grid of values over which to evaluate the posterior density 

Specifies a transformed grid of values over which to evaluate the posterior density 

An alias for the SFACTOR= option 

Writes a note to the log after generating the sample 

Specifies the bounding constant for rejection sampling 

Specifies the number of posterior samples to generate 

Specifies the number of posterior evaluations 

Creates an output data set containing the sample from the posterior density 

Creates an output data set from the grid evaluations 

Creates an output data set from the transformed grid evaluations 

Displays the search used to determine the parameters for the inverted gamma densities 

Displays the transformation of the variance components 

Specifies an integer used to start the pseudorandom number generator 

Adjusts the search range of the transformed parameters 

Inputs the transformation used by the sampling algorithm 

Specifies the algorithm that determines the transformation of the covariance parameters 

An alias for the LOGNOTE= option 
The distribution argument in the PRIOR statement determines the prior density for the variance component parameters of your mixed model. Valid values are as follows.