geweke:92 proposes a convergence diagnostic based on standard time-series methods. The test is appropriate for use with single chains when convergence of the mean of some function of the sampled variables is of interest. For each (function of the) variable, the chain is divided into 2 ``windows'' containing the first (CODA default is 10%) and the last (CODA default is 50%) of the iterates. If the whole chain is stationary, the means of the values early and late in the sequence should be similar. Geweke's approach involves calculation of the sample mean and asymptotic variance in each window, the latter being determined by spectral density estimation. His convergence diagnostic Z is the difference between these 2 means divided by the asymptotic standard error of their difference. As the chain length , the sampling distribution of if the chain has converged. Hence values of Z which fall in the extreme tails of a standard normal distribution suggest that the chain was not fully converged early on (i.e. during the 1st window).
Selecting Geweke (option 1:) from the CODA Diagnostics Menu produces the following output for the line example:
GEWEKE CONVERGENCE DIAGNOSTIC (Z-score): ========================================
Iterations used = 1:200 Thinning interval = 1 Sample size per chain = 200
Fraction in 1st window = 0.1 Fraction in 2nd window = 0.5
-+----------+----------------------+- | VARIABLE | line1 line2 | | ======== | ===== ===== | | | | | alpha | 4.540 -0.488 | | beta | -2.370 -1.100 | | sigma | 5.270 -0.680 | -+----------+----------------------+-
The results for chain line2 provide no evidence against convergence for each variable (although the fact that the Z-scores could be reasonably thought to arise from an N(0, 1) distribution does not prove convergence). However, the values of Z for each monitored variable in chain line1 are rather extreme, suggesting that the first 10% of the samples do not arise from the same distribution as do the last 50%. By discarding the first 20 (10%) iterations of chain line1 and re-computing Geweke's diagnostic, we could next test whether there is any evidence that iterations 21-200 have not converged. If the resulting Z-scores were still extreme, a further 10% of iterations could be discarded and so on.