The FMM Procedure

BAYES Statement

BAYES bayes-options;

The BAYES statement requests that the parameters of the model be estimated by Markov chain Monte Carlo sampling techniques. The FMM procedure can use maximum likelihood to estimate the parameters of all models that are supported by the procedure. Bayesian estimation is available for only a subset of these models.

In Bayesian analysis, it is essential to examine the convergence of the Markov chains before you proceed with posterior inference. With ODS Graphics turned on, the FMM procedure produces graphics at the end of the procedure output; you can visually examine the convergence of the chain in these graphics. You cannot base inferences on a Markov chain that has not converged.

The output produced for a Bayesian analysis is markedly different from that for a frequentist (maximum likelihood) analysis for the following reasons:

Parameter estimates do not have the same interpretation in the two analyses. Parameters are fixed unknown constants in the frequentist context and random variables in a Bayesian analysis.
The results of a Bayesian analysis are summarized through chain diagnostics and posterior summary statistics and intervals.
The FMM procedure samples the mixing probabilities in Bayesian models directly, rather than mapping them onto a logistic (or other) scale.

The FMM procedure applies highly specialized sampling algorithms in Bayesian models. For single-component models without effects, a conjugate sampling algorithm is used where possible. For models in the exponential family that contain effects, the sampling algorithm is based on Gamerman (1997). For the normal and t distributions, a conjugate sampler is the default sampling algorithm for models with and without effects. In multi-component models, the sampling algorithm is based on latent variable sampling through data augmentation (Frühwirth-Schnatter, 2006) and the Gamerman or conjugate sampler. Because of this specialization, the options for controlling the prior distributions of the parameters are limited.

Table 39.3 summarizes the bayes-options available in the BAYES statement. The full assortment of options is then described in alphabetical order.

Table 39.3: BAYES Statement Options

Option	Description
Options Related to Sampling
INITIAL=	Specifies how to construct initial values
NBI=	Specifies the number of burn-in samples
NMC=	Specifies the number of samples after burn-in
METROPOLIS	Forces a Metropolis-Hastings sampling algorithm even if conjugate sampling is possible
OUTPOST=	Generates a data set that contains the posterior estimates
THIN=	Controls the thinning of the Markov chain
Specification of Prior Information
MIXPRIORPARMS	Specifies the prior parameters for the Dirichlet distribution of the mixing probabilities
BETAPRIORPARMS=	Specifies the parameters of the normal prior distribution for individual parameters in the $\bbeta$ vector
MUPRIORPARMS=	Specifies the parameters of the prior distribution for the means in homogeneous mixtures without effects
PHIPRIORPARMS=	Specifies the parameters of the inverse gamma prior distribution for the scale parameters in homogeneous mixtures
PRIOROPTIONS	Specifies additional options used in the determination of the prior distribution
Posterior Summary Statistics and Convergence Diagnostics
DIAGNOSTICS=	Displays convergence diagnostics for the Markov chain
STATISTICS	Displays posterior summary information for the Markov chain
Other Options
ESTIMATE=	Specifies which estimate is used for the computation of OUTPUT statistics and graphics
TIMEINC=	Specifies the time interval to report on sampling progress (in seconds)

You can specify the following bayes-options in the BAYES statement.

BETAPRIORPARMS=pair-specification BETAPRIORPARMS(pair-specification …pair-specification)

specifies the parameters for the normal prior distribution of the parameters that are associated with model effects ( $\beta$ s). The pair-specification is of the form $(a,b)$ , and the values a and b are the mean and variance of the normal distribution, respectively. This option overrides the PRIOROPTIONS option.

The form of the BETAPRIORPARMS with an equal sign and a single pair is used to specify one pair of prior parameters that applies to all components in the mixture. In the following example, the two intercepts and the two regression coefficients all have a $N(0,100)$ prior distribution:

proc fmm;
   model y = x / k=2;
   bayes betapriorparms=(0,100);
run;

You can also provide a list of pairs to specify different sets of prior parameters for the various regression parameters and components. For example:

proc fmm;
   model y = x/ k=2;
   bayes betapriorparms( (0,10) (0,20) (.,.) (3,100) );
run;

The simple linear regression in the first component has a $N(0,10)$ prior for the intercept and a $N(0,20)$ prior for the slope. The prior for the intercept in the second component uses the FMM default, whereas the prior for the slope is $N(3,100)$ .

DIAGNOSTICS=ALL | NONE | (keyword-list) DIAG=ALL | NONE | (keyword-list)

controls the computation of diagnostics for the posterior chain. You can request all posterior diagnostics by specifying DIAGNOSTICS=ALL or suppress the computation of posterior diagnostics by specifying DIAGNOSTICS=NONE. The following keywords enable you to select subsets of posterior diagnostics; the default is DIAGNOSTICS=(AUTOCORR).

AUTOCORR <(LAGS= numeric-list)>

computes for each sampled parameter the autocorrelations of lags specified in the LAGS= list. Elements in the list are truncated to integers, and repeated values are removed. If the LAGS= option is not specified, autocorrelations are computed by default for lags 1, 5, 10, and 50. See the section Autocorrelations in Chapter 7: Introduction to Bayesian Analysis Procedures, for details.

ESS

computes an estimate of the effective sample size (Kass et al., 1998), the correlation time, and the efficiency of the chain for each parameter. See the section Effective Sample Size in Chapter 7: Introduction to Bayesian Analysis Procedures, for details.

GEWEKE <(geweke-options)>

computes the Geweke spectral density diagnostics (Geweke, 1992), which are essentially a two-sample t test between the first $f_1$ portion and the last $f_2$ portion of the chain. The default is $f_1=0.1$ and $f_2=0.5$ , but you can choose other fractions by using the following geweke-options:

FRAC1=value: specifies the fraction $f_1$ for the first window.
FRAC2=value: specifies the fraction $f_2$ for the second window.

See the section Geweke Diagnostics in Chapter 7: Introduction to Bayesian Analysis Procedures, for details.

HEIDELBERGER <(Heidel-options)> HEIDEL <(Heidel-options)>

computes the Heidelberger and Welch diagnostic (which consists of a stationarity test and a half-width test) for each variable. The stationary diagnostic test tests the null hypothesis that the posterior samples are generated from a stationary process. If the stationarity test is passed, a half-width test is then carried out. See the section Heidelberger and Welch Diagnostics in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details.

These diagnostics are not performed by default. You can specify the DIAGNOSTICS=HEIDELBERGER option to request these diagnostics, and you can also specify suboptions, such as DIAGNOSTICS=HEIDELBERGER(EPS=0.05), as follows:

SALPHA=value: specifies the $\alpha$ level $(0<\alpha <1)$ for the stationarity test. By default, SALPHA=0.05.
HALPHA=value: specifies the $\alpha$ level $(0<\alpha <1)$ for the half-width test. By default, HALPHA=0.05.
EPS=value: specifies a small positive number $\epsilon$ such that if the half-width is less than $\epsilon$ times the sample mean of the retaining iterates, the half-width test is passed. By default, EPS=0.1.

MCERROR MCSE

computes an estimate of the Monte Carlo standard error for each sampled parameter. See the section Standard Error of the Mean Estimate in Chapter 7: Introduction to Bayesian Analysis Procedures, for details.

MAXLAG=n

specifies the largest lag used in computing the effective sample size and the Monte Carlo standard error. Specifying this option implies the ESS and MCERROR options. The default is MAXLAG=250.

RAFTERY <(Raftery-options)> RL <(Raftery-options)>

computes the Raftery and Lewis diagnostics, which evaluate the accuracy of the estimated quantile ( $\hat{\theta }_ Q$ for a given Q $\in (0,1)$ ) of a chain. $\hat{\theta }_ Q$ can achieve any degree of accuracy when the chain is allowed to run for a long time. The algorithm stops when the estimated probability $\hat{P}_ Q=\mr{Pr}(\theta \leq \hat{\theta }_ Q)$ reaches within $\pm \mbox{R}$ of the value Q with probability S; that is, $\mr{Pr}(\mbox{Q}-\mbox{R} \leq \hat{\mbox{P}}_ Q \leq \mbox{Q}+\mbox{R})=\mbox{S}$ . See the section Raftery and Lewis Diagnostics in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The Raftery-options enable you to specify Q, R, S, and a precision level $\epsilon$ for a stationary test.

These diagnostics are not performed by default. You can specify the DIAGNOSTICS=RAFERTY option to request these diagnostics, and you can also specify suboptions, such as DIAGNOSTICS=RAFERTY(QUANTILE=0.05), as follows:

QUANTILE=value Q=value: specifies the order (a value between 0 and 1) of the quantile of interest. By default, QUANTILE=0.025.
ACCURACY=value R=value: specifies a small positive number as the margin of error for measuring the accuracy of estimation of the quantile. By default, ACCURACY=0.005.
PROB=value S=value: specifies the probability of attaining the accuracy of the estimation of the quantile. By default, PROB=0.95.
EPS=value: specifies the tolerance level (a small positive number between 0 and 1) for the stationary test. By default, EPS=0.001.

MIXPRIORPARMS=K MIXPRIORPARMS(value-list)

specifies the parameters used in constructing the Dirichlet prior distribution for the mixing parameters. If you specify MIXPRIORPARMS=K, the parameters of the k-dimensional Dirichlet distribution are a vector that contains the number of components in the model (k), whatever that might be. You can specify an explicit list of parameters in value-list. If the MIXPRIORPARMS option is not specified, the default Dirichlet parameter vector is a vector of length k of ones. This results in a uniform prior over the unit simplex; for k=2, this is the uniform distribution. See the section Prior Distributions for the distribution function of the Dirichlet as used by the FMM procedure.

ESTIMATE=MEAN | MAP

determines which overall estimate is used, based on the posterior sample, in the computation of OUTPUT statistics and certain ODS graphics. By default, the arithmetic average of the (thinned) posterior sample is used. If you specify ESTIMATE=MAP, the parameter vector is used that corresponds to the maximum log posterior density in the posterior sample. In any event, a message is written to the SAS log if postprocessing results depend on a summary estimate of the posterior sample.

INITIAL=DATA | MLE | MODE | RANDOM

determines how initial values for the Markov chain are obtained. The default when a conjugate sampler is used is INITIAL=DATA, in which case the FMM procedure uses the same algorithm to obtain data-dependent starting values as it uses for maximum likelihood estimation. If no conjugate sampler is available or if you use the METROPOLIS option to explicitly request that it not be used, then the default is INITIAL=MLE, in which case the maximum likelihood estimates are used as the initial values. If the maximum likelihood optimization fails, the FMM procedure switches to the default INITIAL=DATA.

The options INITIAL=MODE and INITIAL=RANDOM use the mode and random draws from the prior distribution, respectively, to obtain initial values. If the mode does not exist or if it falls on the boundary of the parameter space, the prior mean is used instead.

METROPOLIS

requests that the FMM procedure use the Metropolis-Hastings sampling algorithm based on Gamerman (1997), even in situations where a conjugate sampler is available.

MUPRIORPARMS=pair-specification MUPRIORPARMS(pair-specification …pair-specification)

specifies the parameters for the means in homogeneous mixtures without regression coefficients. The pair-specification is of the form $(a,b)$ , where a and b are the two parameters of the prior distribution, optionally delimited with a comma. The actual distribution of the parameter is implied by the distribution selected in the MODEL statement. For example, it is a normal distribution for a mixture of normals, a gamma distribution for a mixture of Poisson variables, a beta distribution for a mixture of binary variables, and an inverse gamma distribution for a mixture of exponential variables. This option overrides the PRIOROPTIONS option.

The parameters correspond as follows:

Beta:: The parameters correspond to the $\alpha$ and $\beta$ parameters of the beta prior distribution such that its mean is $\mu = \alpha /(\alpha +\beta )$ and its variance is $\mu (1-\mu )/(\alpha +\beta +1)$ .
Normal:: The parameters correspond to the mean and variance of the normal prior distribution.
Gamma:: The parameters correspond to the $\alpha$ and $\beta$ parameters of the gamma prior distribution such that its mean is $\alpha /\beta$ and its variance is $\alpha /\beta ^2$ .
Inverse gamma:: The parameters correspond to the $\alpha$ and $\beta$ parameters of the inverse gamma prior distribution such that its mean is $\mu = \beta /(\alpha -1)$ and its variance is $\mu ^2/(\alpha -2)$ .

The two techniques for specifying the prior parameters with the MUPRIORPARMS option are as follows:

Specify an equal sign and a single pair of values:

proc fmm seed=12345;
   model y = / k=2;
   bayes mupriorparms=(0,50);
run;

Specify a list of parameter pairs within parentheses:

proc fmm seed=12345;
   model y = / k=2;
   bayes mupriorparms( (.,.) (1.4,10.5));
run;

If you specify an invalid value (outside of the parameter space for the prior distribution), the FMM procedure chooses the default value and writes a message to the SAS log. If you want to use the default values for a particular parameter, you can also specify missing values in the pair-specification. For example, the preceding list specification assigns default values for the first component and uses the values 1.4 and 10.5 for the mean and variance of the normal prior distribution in the second component. The first example assigns a $N(0,50)$ prior distribution to the means in both components.

NBI=n

specifies the number of burn-in samples. During the burn-in phase, chains are not saved. The default is NBI=2000.

NMC=n SAMPLE=n

specifies the number of Monte Carlo samples after the burn-in. Samples after the burn-in phase are saved unless they are thinned with the THIN= option. The default is NMC=10000.

OUTPOST<(outpost-options)>=data-set

requests that the posterior sample be saved to a SAS data set. In addition to variables that contain log likelihood and log posterior values, the OUTPOST data set contains variables for the parameters. The variable names for the parameters are generic (Parm_1, Parm_2, $\cdots$ , Parm_p). The labels of the parameters are descriptive and correspond to the "Parameter Mapping" table that is produced when the OUTPOST= option is in effect.

You can specify the following outpost-options in parentheses:

LOGPRIOR: adds the value of the log prior distribution to the data set.
NONSINGULAR | NONSING | COMPRESS: eliminates parameters that correspond to singular columns in the design matrix (and were not sampled) from the posterior data set. This is the default.
SINGULAR | SING: adds columns of zeros to the data set in positions that correspond to singularities in the model or to parameters that were not sampled for other reasons. By default, these columns of zeros are not written to the posterior data set.

PHIPRIORPARMS=pair-specification PHIPRIORPARMS( pair-specification …pair-specification)

specifies the parameters for the inverse gamma prior distribution of the scale parameters ( $\phi$ ’s) in the model. The pair-specification is of the form $(a, b)$ , and the values are chosen such that the prior distribution has mean $\mu = b/(a-1)$ and variance $\mu ^2/(a-2)$ .

The form of the PHIPRIORPARMS with an equal sign and a single pair is used to specify one pair of prior parameters that applies to all components in the mixture. For example:

proc fmm seed=12345;
   model y = / k=2;
   bayes phipriorparms=(2.001,1.001);
run;

The form with a list of pairs is used to specify different prior parameters for the scale parameters in different components. For example:

proc fmm seed=12345;
   model y = / k=2;
   bayes phipriorparms( (.,1.001)  (3.001,2.001) );
run;

If you specify an invalid value (outside of the parameter space for the prior distribution), the FMM procedure chooses the default value and writes a message to the SAS log. If you want to use the default values for a particular parameter, you can also specify missing values in the pair-specification. For example, the preceding list specification assigns default values for the first component a prior parameter and uses the value 1.001 for the b prior parameter. The second pair assigns 3.001 and 2.001 for the a and b prior parameters, respectively.

PRIOROPTIONS <=>(prior-options) PRIOROPTS <=>(prior-options)

specifies options related to the construction of the prior distribution and the choice of their parameters. Some prior-options apply only in particular models. The BETAPRIORPARMS= and MUPRIORPARMS= options override this option.

You can specify the following prior-options:

CONDITIONAL | COND

chooses a conditional prior specification for the homogeneous normal and t distribution response components. The default prior specification in these models is an independence prior where the mean of the hth component has prior $\mu _ h \sim N(a,b)$ . The conditional prior is characterized by $\mu _ h \sim N(a, \sigma ^2_ h/b)$ .

DEPENDENT | DEP

chooses a data-dependent prior for the homogeneous models without effects. The prior parameters a and b are chosen as follows, based on the distribution in the MODEL statement:

Binary and binomial:

$a = \bar{y}/(1-\bar{y})$ , b=1, and the prior distribution for the success probability is $\mr{beta}(a,b)$ .

Poisson:

$a = 1$ , $b = 1/\, \bar{y}$ , and the prior distribution for $\mu$ is $\mr{gamma}(a,b)$ . See Frühwirth-Schnatter (2006, p. 280) and Viallefont, Richardson, and Greene (2002).

Exponential:

$a = 3$ , $b=2\bar{y}$ , and the prior distribution for $\mu$ is inverse gamma with parameters a and b.

Normal and t:

Under the default independence prior, the prior distribution for $\mu$ is $N(\bar{y}, f s^2)$ where f is the variance factor from the VAR= option and

$s^2 = \frac{1}{n} \sum _{i=1}^ n \left( y_ i - \bar{y}\right)^2$

Under the default conditional prior specification, the prior for $\mu _ h$ is $N(a,\sigma ^2_ h/b)$ where $a = \bar{y}$ and $b = 2.6/(\max \{ y\} -\min \{ y\} )$ . The prior for the scale parameter is inverse gamma with parameters 1.28 and $0.36s^2$ . For further details, see Raftery (1996) and Frühwirth-Schnatter (2006, p. 179).

VAR=f

specifies the variance for normal prior distributions. The default is VAR=1000. This factor is used, for example, in determining the prior variance of regression coefficients or in determining the prior variance of means in homogeneous mixtures of t or normal distributions (unless a data-dependent prior is used).

MLE<=r>

specifies that the prior distribution for regression variables be based on a multivariate normal distribution centered at the MLEs and whose dispersion is a multiple r of the asymptotic MLE covariance matrix. The default is MLE=10. In other words, if you specify PRIOROPTIONS(MLE), the FMM procedure chooses the prior distribution for the regression variables as $N(\widehat{\beta },10\mr{Var}[\widehat{\beta }])$ where $\widehat{\beta }$ is the vector of maximum likelihood estimates. The prior for the scale parameter is inverse gamma with parameters 1.28 and $0.36s^2$ where

$s^2 = \frac{1}{n} \sum _{i=1}^ n \left( y_ i - \bar{y}\right)^2$

For further details, see Raftery (1996) and Frühwirth-Schnatter (2006, p. 179). If you specify PRIOROPTIONS(MLE) for the regression parameters, then the data-dependent prior is used for the scale parameter; see the PRIOROPTIONS(DEPENDENT) option above.

The MLE option is not available for mixture models in which the parameters are estimated directly on the data scale, such as homogeneous mixture models or mixtures of distributions without model effects for which a conjugate sampler is available. By using the METROPOLIS option, you can always force the FMM procedure to abandon a conjugate sampler in favor of a Metropolis-Hastings sampling algorithm to which the MLE option applies.

controls the number of posterior statistics produced. Specifying STATISTICS=ALL is equivalent to specifying STATISTICS=(SUMMARY INTERVAL). To suppress the computation of posterior statistics, specify STATISTICS=NONE. The default is STATISTICS=(SUMMARY INTERVAL). See the section Summary Statistics in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details.

The global-options include the following:

ALPHA=numeric-list: controls the coverage levels of the equal-tail credible intervals and the credible intervals of highest posterior density (HPD) credible intervals. The ALPHA= values must be between 0 and 1. Each ALPHA= value produces a pair of $100(1-\alpha )$ % equal-tail and HPD credible intervals for each sampled parameter. The default is ALPHA=0.05, which results in 95% credible intervals for the parameters.
PERCENT=numeric-list: requests the percentile points of the posterior samples. The values in numeric-list must be greater than or equal to 0 and less than or equal to 100. The default is PERCENT=(25 50 75), which yields for each parameter the 25th, 50th, and 75th percentiles, respectively.

The list of keywords includes the following:

SUMMARY: produces the means, standard deviations, and percentile points for the posterior samples. The default is to produce the 25th, 50th, and 75th percentiles; you can modify this list with the global PERCENT= option.
INTERVAL: produces equal-tail and HPD credible intervals. The default is to produce the 95% equal-tail credible intervals and 95% HPD credible intervals, but you can use the ALPHA= global-option to request credible intervals for any probabilities.

THIN=n THINNING=n

controls the thinning of the Markov chain after the burn-in. Only one in every k samples is used when THIN=k, and if NBI= $n_0$ and NMC=n, the number of samples kept is

$\biggl [ \frac{n_0+n}{k} \biggr ] - \biggl [ \frac{n_0}{k} \biggr ]$

where [a] represents the integer part of the number a. The default is THIN=1—that is, all samples are kept after the burn-in phase.

TIMEINC=n

specifies a time interval in seconds to report progress during the burn-in and sampling phase. The time interval is approximate, because the minimum time interval in which the FMM procedure can respond depends on the multithreading configuration.