Proper transformations of parameters can often improve the mixing in PROC MCMC. You already saw this in Nonlinear Poisson Regression Models, which sampled using the scale of parameters that priors that are strictly positive, such as the gamma priors. This example shows another useful transformation: the logit transformation on parameters that take a uniform prior on [0, 1].
The data set is taken from Sharples (1990). It is used in Chaloner and Brant (1988) and Chaloner (1994) to identify outliers in the data set in a two-level hierarchical model. Congdon (2003) also uses this data set to demonstrates the same technique. This example uses the data set to illustrate how mixing can be improved using transformation and does not address the question of outlier detection as in those papers. The following statements create the data set:
data inputdata; input nobs grp y @@; ind = _n_; datalines; 1 1 24.80 2 1 26.90 3 1 26.65 4 1 30.93 5 1 33.77 6 1 63.31 1 2 23.96 2 2 28.92 3 2 28.19 4 2 26.16 5 2 21.34 6 2 29.46 1 3 18.30 2 3 23.67 3 3 14.47 4 3 24.45 5 3 24.89 6 3 28.95 1 4 51.42 2 4 27.97 3 4 24.76 4 4 26.67 5 4 17.58 6 4 24.29 1 5 34.12 2 5 46.87 3 5 58.59 4 5 38.11 5 5 47.59 6 5 44.67 ;
There are five groups (grp
, ) with six observations (nobs
, ) in each. The two-level hierarchical model is specified as follows:
with the precision parameters related to each other in the following way:
The total number of parameters in this model is eight: , and p.
The following statements fit the model:
ods graphics on; proc mcmc data=inputdata nmc=50000 thin=10 outpost=m1 seed=17 plot=trace; ods select ess tracepanel; parms p; parms tau; parms mu; prior p ~ uniform(0,1); prior tau ~ gamma(shape=0.001,iscale=0.001); prior mu ~ normal(0,prec=0.00000001); beginnodata; taub = tau/p; tauw = taub-tau; endnodata; random theta ~ normal(mu, prec=taub) subject=grp monitor=(theta); model y ~ normal(theta,prec=tauw); run;
The ODS SELECT statement displays the effective sample size table and the trace plots. The ODS GRAPHICS ON statement enables
ODS Graphics. The PROC MCMC statement specifies the usual options for the procedure run and produces trace plots (PLOTS=TRACE
). The three PARMS
statements put three model parameters, p
, tau
, and mu
, in three different blocks. The PRIOR
statements specify the prior distributions, and the programming statements enclosed with the BEGINNODATA
and ENDNODATA
statements calculate the transformation to taub
and tauw
. The RANDOM
statement specifies the random effect, its prior distribution, and the subject variable. The resulting trace plots are shown
in Output 73.20.1, and the effective sample size table is shown in Output 73.20.2.
Output 73.20.1: Trace Plots
Output 73.20.2: Bad Effective Sample Sizes
Effective Sample Sizes | |||
---|---|---|---|
Parameter | ESS | Autocorrelation Time |
Efficiency |
p | 81.3 | 61.5342 | 0.0163 |
tau | 61.2 | 81.7440 | 0.0122 |
mu | 5000.0 | 1.0000 | 1.0000 |
theta_1 | 4839.9 | 1.0331 | 0.9680 |
theta_2 | 2739.7 | 1.8250 | 0.5479 |
theta_3 | 1346.6 | 3.7130 | 0.2693 |
theta_4 | 4897.5 | 1.0209 | 0.9795 |
theta_5 | 338.1 | 14.7866 | 0.0676 |
The trace plots show that most parameters have relatively good mixing. Two exceptions appear to be p and . The trace plot of p shows a slow periodic movement. The parameter does not have good mixing either. When the values are close to zero, the chain stays there for periods of time. An inspection of the effective sample sizes table reveals the same conclusion: p and have much smaller ESSs than the rest of the parameters.
A scatter plot of the posterior samples of p and reveals why mixing is bad in these two dimensions. The following statements generate the scatter plot in Output 73.20.3:
title 'Scatter Plot of Parameters on Original Scales'; proc sgplot data=m1; yaxis label = 'p'; xaxis label = 'tau' values=(0 to 0.4 by 0.1); scatter x = tau y = p; run;
Output 73.20.3: Scatter Plot of versus p
The two parameters clearly have a nonlinear relationship. It is not surprising that the Metropolis algorithm does not work well here. The algorithm is designed for cases where the parameters are linearly related with each other.
To improve on mixing, you can sample on the log of , instead of sampling on . The formulation is:
See the section Standard Distributions for the definitions of the gamma and egamma distributions. In addition, you can sample on the logit of p. Note that
is equivalent to
The following statements fit the same model by using transformed parameters:
proc mcmc data=inputdata nmc=50000 thin=10 outpost=m2 seed=17 monitor=(p tau mu) plot=trace; ods select ess tracepanel; parms ltau lgp mu ; prior ltau ~ egamma(shape=0.001,iscale=0.001); prior lgp ~ logistic(0,1); prior mu ~ normal(0,prec=0.00000001); beginnodata; tau = exp(ltau); p = logistic(lgp); taub = tau/p; tauw = taub-tau; endnodata; random theta ~ normal(mu, prec=taub) subject=grp monitor=(theta); model y ~ normal(theta,prec=tauw); run;
The variable lgp
is the logit transformation of p, and ltau
is the log transformation of . The prior for ltau
is egamma
, and the prior for lgp
is logistic
. The TAU and P assignment statements transform the parameters back to their original scales. The rest of the programs remain
unchanged. Trace plots (Output 73.20.4) and effective sample size (Output 73.20.5) both show significant improvements in the mixing for both p and .
Output 73.20.4: Trace Plots after Transformation
Output 73.20.5: Effective Sample Sizes after Transformation
Effective Sample Sizes | |||
---|---|---|---|
Parameter | ESS | Autocorrelation Time |
Efficiency |
p | 3119.4 | 1.6029 | 0.6239 |
tau | 2588.0 | 1.9320 | 0.5176 |
mu | 5000.0 | 1.0000 | 1.0000 |
theta_1 | 4866.0 | 1.0275 | 0.9732 |
theta_2 | 5244.5 | 0.9534 | 1.0489 |
theta_3 | 5000.0 | 1.0000 | 1.0000 |
theta_4 | 5000.0 | 1.0000 | 1.0000 |
theta_5 | 4054.8 | 1.2331 | 0.8110 |
The following statements generate Output 73.20.6 and Output 73.20.7:
title 'Scatter Plot of Parameters on Transformed Scales'; proc sgplot data=m2; yaxis label = 'logit(p)'; xaxis label = 'log(tau)'; scatter x = ltau y = lgp; run; title 'Scatter Plot of Parameters on Original Scales'; proc sgplot data=m2; yaxis label = 'p'; xaxis label = 'tau' values=(0 to 5.0 by 1); scatter x = tau y = p; run; ods graphics off;
Output 73.20.6: Scatter Plot of versus , After Transformation
Output 73.20.7: Scatter Plot of versus p, After Transformation
The scatter plot of versus shows a linear relationship between the two transformed parameters, and this explains the improvement in mixing. In addition,
the transformations also help the Markov chain better explore in the original parameter space. Output 73.20.7 shows a scatter plot of versus p. The plot is similar to Output 73.20.3. However, note that tau
has a far longer tail in Output 73.20.7, extending all the way to 5 as opposed to 0.15 in Output 73.20.3. This means that the second Markov chain can explore this dimension of the parameter more efficiently, and as a result, you
are able to draw more precise inference with an equal number of simulations.