This example illustrates how you can obtain random samples from a known function. The target distributions are the normal distribution (a standard distribution) and a mixture of the normal distributions (a nonstandard distribution). For more information, see the sections Standard Distributions and Specifying a New Distribution). This example also shows how you can use PROC MCMC to estimate an integral (area under a curve). Monte Carlo simulation is data-independent; hence, you do not need an input data set from which to draw random samples from the desired distribution.
When you run a simulation without an input data set, the posterior distribution is the same as the prior distribution. Hence, if you want to generate samples from a distribution, you declare the distribution in the PRIOR statement and set the likelihood function to a constant. Although there is no contribution from any data set variable to the likelihood calculation, you still must specify a data set and the MODEL statement needs a distribution. You can input an empty data set and use the GENERAL function to provide a flat likelihood. The following statements generate 10,000 samples from a standard normal distribution:
title 'Simulating Samples from a Normal Density'; data x; run; ods graphics on; proc mcmc data=x outpost=simout seed=23 nmc=10000 diagnostics=none; ods exclude nobs; parm alpha 0; prior alpha ~ normal(0, sd=1); model general(0); run;
The ODS GRAPHICS ON statement enables ODS Graphics. The PROC MCMC statement specifies the input and output data sets, a random number seed, and the size of the simulation sample. The STATISTICS= option displays only the summary and interval statistics. The ODS EXCLUDE statement excludes the display of the NObs table. PROC MCMC draws independent samples from the normal distribution directly (see Output 73.1.1). Therefore, the simulation does not require any tuning, and PROC MCMC omits the default burn-in phrase.
Output 73.1.1: Parameters Information
The summary statistics (Output 73.1.2) are what you would expect from a standard normal distribution.
Output 73.1.2: MCMC Summary and Interval Statistics from a Normal Target Distribution
The trace plot (Output 73.1.3) shows perfect mixing with no autocorrelation in the lag plot. This is expected because these are independent draws.
Output 73.1.3: Diagnostics Plots for
You can overlay the estimated kernel density with the true density to visually compare the densities, as displayed in Output 73.1.4. To create the kernel comparison plot, you first call PROC KDE (see Chapter 66: The KDE Procedure) to obtain a kernel density estimate of the posterior density on alpha
. Then you evaluate a grid of alpha
values on a normal density. The following statements evaluate kernel density and compute the corresponding normal density:
proc kde data=simout; ods exclude inputs controls; univar alpha /out=sample; run; data den; set sample; alpha = value; true = pdf('normal', alpha, 0, 1); keep alpha density true; run;
Next you plot the two curves on top of each other by using PROC SGPLOT (see Chapter 21: Statistical Graphics Using ODS) as follows:
proc sgplot data=den; yaxis label="Density"; series y=density x=alpha / legendlabel = "MCMC Kernel"; series y=true x=alpha / legendlabel = "True Density"; discretelegend; run;
Output 73.1.4 shows the result. You can see that the kernel estimate and the true density are very similar to each other.
Output 73.1.4: Estimated Density versus the True Density
In programs that do not involve any data set variables, PROC MCMC samples directly from the (joint) prior distributions of the parameters. The modification makes the sampling from a known distribution more efficient and more precise. For example, you can write simple programs, such as the following macro, to understand different aspects of a prior distribution of interest, such as its moments, intervals, shape, spread, and so on:
%macro density(dist=, seed=0); %let savenote = %sysfunc(getoption(notes)); options nonotes; title "&dist distribution."; data _a; run; ods select densitypanel postsumint; proc mcmc data=_a nmc=10000 diag=none nologdist plots=density seed=&seed; parms alpha; prior alpha ~ &dist; model general(0); run; proc datasets nolist; delete _a; run; options &savenote; %mend; %density(dist=beta(4, 12), seed=1);
The macro %density
creates an empty data set, invokes PROC MCMC, draws 10,000 samples from a beta(4, 12) distribution, displays summary and
interval statistics, and generates a kernel density plot. Summary and interval statistics from the beta distribution are displayed
in Output 73.1.5.
Output 73.1.5: Beta Distribution Statistics
The distribution is displayed in Output 73.1.6.
Output 73.1.6: Density Plot
One advantage of MCMC methods is to estimate any integral under the curve of a target distribution. This can be done fairly easily using the MCMC procedure. Suppose you are interested in estimating the following cumulative probability:
To estimate this integral, PROC MCMC draws samples from the distribution and counts the portion of the simulated values that fall within the desired range of [0, 1.3]. This becomes a Monte Carlo estimate of the integral. The following statements simulate samples from a standard normal distribution and estimate the integral:
proc mcmc data=x outpost=simout seed=23 nmc=10000 nologdist monitor=(int) diagnostics=none; ods select postsumint; parm alpha 0; prior alpha ~ normal(0, sd=1); int = (0 <= alpha <= 1.3); model general(0); run;
The ODS SELECT statement displays the posterior summary statistics table. The MONITOR=
option outputs analysis on the variable int
(the integral estimate). The STATISTICS=
option computes the summary statistics. The NOLOGDIST
option omits the calculation of the log of the prior distribution at each iteration, shortening the simulation time^{[37]}. The INT assignment statement sets int
to be 1 if the simulated alpha
value falls between 0 and 1.3, and 0 otherwise. PROC MCMC supports the usage of the IF-ELSE logical control if you need to
account for more complex conditions. Output 73.1.7 displays the estimated integral value:
Output 73.1.7: Monte Carlo Integral from a Normal Distribution
In this simulation, 4079 samples fall between 0 and 1.3, making the expected probability 0.4079. In this example, you can verify the actual cumulative probability by calling the CDF function in the DATA step:
data _null_; int = cdf("normal", 1.3, 0, 1) - cdf("normal", 0, 0, 1); put int=; run;
The value is 0.4032.
Suppose you are interested in generating samples from a three-component mixture of normal distributions, with the density specified as follows:
You can either specify the distribution directly or use a latent variable approach to generate samples from the normal mixture.
To specify the normal mixture density directly in PROC MCMC, you need to construct the function because the normal mixture distribution is not one of the standard distributions that PROC MCMC supports. The following statements generate random samples from the normal mixture density:
title 'Simulating Samples from a Mixture of Normal Densities'; data x; run; proc mcmc data=x outpost=simout seed=1234 nmc=30000; ods select TADpanel; parm alpha 0.3; lp = logpdf('normalmix', alpha, 3, 0.3, 0.4, 0.3, -3, 2, 10, 2, 1, 4); prior alpha ~ general(lp); model general(0); run;
The ODS SELECT statement displays the diagnostic plots. All other tables, such as the NObs
tables, are excluded. The PROC MCMC statement uses the input data set X
, saves output to the Simout
data set, sets a random number seed, and draws 30,000 samples.
The LP assignment statement evaluates the log density of alpha
at the mixture density, using the SAS function LOGPDF. The number 3 after alpha
in the LOGPDF function indicates that the density is a three-component normal mixture. The following three numbers, 0.3,
0.4, and 0.3, are the weights in the mixture; –3, 2, and 10 are the means; 2, 1, and 4 are the standard deviations. The PRIOR
statement assigns this log density function to alpha
as its prior. Note that the GENERAL
function interprets the density on the log scale, and not the original scale–you must use the LOGPDF function, not the PDF
function. Output 73.1.8 displays the results. The kernel density clearly shows three modes.
Output 73.1.8: Plots of Posterior Samples from a Mixture Normal Distribution
Alternatively, the normal mixture distribution can also decomposed into a marginal distribution for the component (call it Z) and a conditional model of the response variable Y given Z, as
where K is the total number of mixture components, z is the component indicator, and and are the model parameters for the zth component.
Starting with SAS/STAT 13.2, PROC MCMC supports a categorical distribution that can be used to model the discrete random variable for components. You can use PRIOR statements to specify a normal mixture distribution and generate samples accordingly:
proc mcmc data=x outpost=simout_m seed=1234 nmc=30000; array p[3] (0.3 0.4 0.3); array mu[3] (-3 2 10); array sd[3] (2 1 4); parm z alpha; prior z ~ table(p); prior alpha ~ normal(mu[z], sd=sd[z]); model general(0); run;
The ARRAY
statements define one array p
for the mixture weights, one array mu
for the means of the normal distributions, and one array sd
for the corresponding standard deviations. The PRIOR
statements specify a categorical prior on the parameter z
and a conditional normal prior on alpha
. The mean and standard deviation of the alpha
parameter depend on the component indicator z
. No output is created.
You can use the following set of statements, which are similar to the previous example, to overlay the estimated kernel density with the true density. The comparison is shown in Output 73.1.9.
proc kde data=simout_m; ods exclude inputs controls; univar alpha /out=sample; run; data den; set sample; alpha = value; true = pdf('normalmix', alpha, 3, 0.3, 0.4, 0.3, -3, 2, 10, 2, 1, 4); keep alpha density true; run; proc sgplot data=den; yaxis label="Density"; series y=density x=alpha / legendlabel = "MCMC Kernel - Latent Variable Approach"; series y=true x=alpha / legendlabel = "True Density"; discretelegend; run;
Output 73.1.9: Estimated Density (Latent Variable Approach) versus the True Density