Convergence and Efficiency Evaluation

Mauricio Garnier-Villarreal

Introduction

When Bayesian models are estimated with a Markov-Chain Monte Carlo (MCMC) sampler, the model estimation doesn’t stop when it has achieved some convergence criteria. It will run as long as desired (determined by the burnin and sample arguments), and then you need to evaluate the convergence and efficiency of the estimated posterior distributions. You should only analyze the results if convergence has been achieved, as judged by the metrics described below.

For this example we will use the Industrialization and Political Democracy example (Bollen 1989).

model <- '
  # latent variable definitions
     ind60 =~ x1 + x2 + x3
     dem60 =~ a*y1 + b*y2 + c*y3 + d*y4
     dem65 =~ a*y5 + b*y6 + c*y7 + d*y8

  # regressions
    dem60 ~ ind60
    dem65 ~ ind60 + dem60

  # residual correlations
    y1 ~~ y5
    y2 ~~ y4 + y6
    y3 ~~ y7
    y4 ~~ y8
    y6 ~~ y8
'

fit <- bsem(model, data=PoliticalDemocracy,
            std.lv=T, meanstructure=T, n.chains=3,
            burnin=500, sample=1000)

Convergence

The primary convergence diagnostic is $\hat{R}$, which compares the between- and within-chain samples of model parameters and other univariate quantities of interest (Vehtari et al. 2021). If chains have not mixed well (ie, the between- and within-chain estimates don’t agree), $\hat{R}$ is larger than 1. We recommend running at least three chains by default and only using the posterior samples if $\hat{R} < 1.05$ for all the parameters.

blavaan presents the $\hat{R}$ reported by the underlying MCMC program, either Stan or JAGS (Stan by default). We can obtain the $\hat{R}$ from the summary() function, and we can also extract it with the blavInspect() function

blavInspect(fit, "rhat")

##   ind60=~x1   ind60=~x2   ind60=~x3           a           b           c 
##   0.9998329   1.0009269   1.0012448   1.0002433   0.9996886   0.9997942 
##           d           a           b           c           d dem60~ind60 
##   0.9996935   1.0002433   0.9996886   0.9997942   0.9996935   1.0004682 
## dem65~ind60 dem65~dem60      y1~~y5      y2~~y4      y2~~y6      y3~~y7 
##   0.9994272   0.9995837   1.0001069   0.9996087   0.9993587   0.9993821 
##      y4~~y8      y6~~y8      x1~~x1      x2~~x2      x3~~x3      y1~~y1 
##   1.0004460   0.9993719   1.0002074   0.9992664   0.9996132   0.9999948 
##      y2~~y2      y3~~y3      y4~~y4      y5~~y5      y6~~y6      y7~~y7 
##   0.9996140   0.9992549   0.9998172   1.0000120   0.9993669   0.9993283 
##      y8~~y8        x1~1        x2~1        x3~1        y1~1        y2~1 
##   0.9995331   1.0062831   1.0058931   1.0048471   1.0032289   1.0025551 
##        y3~1        y4~1        y5~1        y6~1        y7~1        y8~1 
##   1.0006281   1.0035112   1.0024221   1.0042214   1.0050368   1.0028954

With large models it can be cumbersome to look over all of these entries. We can instead find the largest $\hat{R}$ to see if they are all less than $1.05$

max(blavInspect(fit, "psrf"))

## [1] 1.006283

If all $\hat{R} < 1.05$ then we can establish that the MCMC chains have converged to a stable solution. If the model has not converged, you might increase the number of burnin iterations

fit <- bsem(model, data=PoliticalDemocracy,
            std.lv=T, meanstructure=T, n.chains=3,
            burnin=1000, sample=1000)

and/or change the model priors with the dpriors() function. These address issues where the model failed to converge due to needing more iterations or due to a model misspecification (such as bad priors). As a rule of thumb, we seldom see a model require more than 1,000 burnin samples in Stan. If your model is not converging after 1,000 burnin samples, it is likely that the default prior distributions clash with your data. This can happen, e.g., if your variables contain values in the 100s or 1000s.

Efficiency

We should also evaluate the efficiency of the posterior samples. Effective sample size (ESS) is a useful measure for sampling efficiency, and is well defined even if the chains do not have finite mean or variance (Vehtari et al. 2021).

In short, the posterior samples produced by MCMC are autocorrelated. This means that, if you draw 500 posterior samples, you do not have 500 independent pieces of information about the posterior distribution, because the samples are autocorlated. The ESS metric is like a currency conversion, telling you how much your autocorrelated samples are worth if we were to convert them to independent samples. In blavaan we can print it from the summary function with the neff argument

summary(fit, neff=T)

We can also extract only those with the blavInspect() function

blavInspect(fit, "neff")

##   ind60=~x1   ind60=~x2   ind60=~x3           a           b           c 
##    1788.407    1643.328    1871.459    2173.497    2498.604    2204.155 
##           d           a           b           c           d dem60~ind60 
##    2119.537    2173.497    2498.604    2204.155    2119.537    2678.910 
## dem65~ind60 dem65~dem60      y1~~y5      y2~~y4      y2~~y6      y3~~y7 
##    3760.840    3330.003    2711.883    2571.530    3307.739    2654.652 
##      y4~~y8      y6~~y8      x1~~x1      x2~~x2      x3~~x3      y1~~y1 
##    2717.897    1776.845    2168.178    1809.031    3810.470    2942.691 
##      y2~~y2      y3~~y3      y4~~y4      y5~~y5      y6~~y6      y7~~y7 
##    3547.369    3427.482    2621.538    2290.928    2491.826    2377.952 
##      y8~~y8        x1~1        x2~1        x3~1        y1~1        y2~1 
##    1805.490    1278.027    1223.059    1368.768    1422.557    1651.289 
##        y3~1        y4~1        y5~1        y6~1        y7~1        y8~1 
##    1565.144    1332.832    1329.148    1415.159    1332.593    1329.845

ESS is a sample size, so it should be at least 100 (optimally, much more than 100) times the number of chains in order to be reliable and to indicate that estimates of the posterior quantiles are reliable. In this example, because we have 3 chains, we would want to see at least neff=300 for every parameter.

And we can easily find the lowest ESS with the min() function:

min(blavInspect(fit, "neff"))

## [1] 1223.059

References

Bollen, Kenneth A. 1989. Structural Equations with Latent Variables. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc.

Vehtari, Aki, Andrew Gelman, Daniel Simpson, Bob Carpenter, and Paul-Christian Bürkner. 2021. “Rank-Normalization, Folding, and Localization: An Improved $\widehat{R}$ for Assessing Convergence of MCMC (with Discussion).” Bayesian Analysis 16 (2): 667–718. https://doi.org/10.1214/20-BA1221.