The Geweke diagnostic and the Heidelberg and Welch diagnostic are reported. These two convergence diagnostics are calculated based on only a single MCMC chain. Both diagnostics require a single chain and may be applied with any MCMC method. The functions geweke.diag, heidel.diag in coda package is used to compute this diagnostic.
Geweke's convergence diagnostic is calculated by taking the difference between the means from the first \(n_A\) iterations and the last \(n_B\) iterations. If the ratios \(n_A/n\) and \(n_B/n\) are fixed and \(nA +nB < n\), then by the central limit theorem, the distribution of this diagnostic approaches a standard normal as \(n\) tends to infinity. In our package, \(n_A= .2*n\) and \(n_B= .5*n\).
The Heidelberg and Welch diagnostic is based on a test statistic to accept or reject the null hypothesis that the Markov chain is from a stationary distribution. The present package reports the stationary test.The convergence test uses the Cramer-von Mises statistic to test for stationary. The test is successively applied on the chain. If the null hypothesis is rejected, the first 10% of the iterations are discarded and the stationarity test repeated. If the stationary test fails again, an additional 10% of the iterations are discarded and the test repeated again. The process continues until 50% of the iterations have been discarded and the test still rejects. In our package, \(eps = 0.1, pvalue = 0.05\) are used as parameters of the function heidel.diag.
conv.diag(x)the object from BANOVA.*
conv.diag returns a list of two diagnostics:
The Geweke diagnostic
The Heidelberg and Welch diagnostic
Plummer, M., Best, N., Cowles, K. and Vines K. (2006) CODA: Convergence Diagnosis and Output Analysis for MCMC, R News, Vol 6, pp. 7-11.
Geweke, J. Evaluating the accuracy of sampling-based approaches to calculating posterior moments, In Bayesian Statistics 4 (ed JM Bernado, JO Berger, AP Dawid and AFM Smith). Clarendon Press, Oxford, UK.
Heidelberger, P. and Welch, PD. (1981) A spectral method for confidence interval generation and run length control in simulations, Comm. ACM. Vol. 24, No.4, pp. 233-245.
Heidelberger, P. and Welch, PD. (1983) Simulation run length control in the presence of an initial transient, Opns Res., Vol.31, No.6, pp. 1109-44.
Schruben, LW. (1982) Detecting initialization bias in simulation experiments, Opns. Res., Vol. 30, No.3, pp. 569-590.
# NOT RUN {
data(goalstudy)
# }
# NOT RUN {
library(rstan)
res1 <- BANOVA.run(bid~progress*prodvar, model_name = "Normal", data = goalstudy,
id = 'id', iter = 100, thin = 1)
conv.diag(res1)
# might need pairs() to confirm the convergence
# }
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