Editing Markov chain Monte Carlo (section)

=== Geweke Diagnostics ===

The Geweke diagnostic examines whether the distribution of samples in the early part of the Markov chain is statistically indistinguishable from the distribution in a later part.<ref>{{Citation |last=Geweke |first=John |title=Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments |date=1992-08-13 |work=Bayesian Statistics 4 |pages=169–194 |editor-last=Bernardo |editor-first=J M |url=https://academic.oup.com/book/54041/chapter/422209572 |access-date=2025-04-29 |publisher=Oxford University PressOxford |language=en |doi=10.1093/oso/9780198522669.003.0010 |isbn=978-0-19-852266-9 |editor2-last=Berger |editor2-first=J O |editor3-last=Dawid |editor3-first=P |editor4-last=Smith |editor4-first=A F M}}</ref> Given a sequence of correlated MCMC samples <math>\{X_1, X_2, \dots, X_n\}</math>, the diagnostic splits the chain into an early segment consisting of the first <math>n_A</math> samples, typically chosen as <math>n_A=0.1n</math> (i.e., the first 10% of the chain), and a late segment consisting of the last <math>n_B</math> samples, typically chosen as <math>n_B=0.5n</math> (i.e., the last 50% of the chain)

Denote the sample means of these segments as:

:<math>
\bar{X}_A = \dfrac{1}{n_A}\sum_{i=1}^{n_A} X_i,\;\;\; \bar{X}_B = \dfrac{1}{n_B}\sum_{i=n-n_B+1}^n X_i
</math>

Since MCMC samples are autocorrelated, a simple comparison of sample means is insufficient. Instead, the difference in means is standardized using an estimator of the spectral density at zero frequency, which accounts for the long-range dependencies in the chain. The test statistic is computed as:

:<math>
Z = \dfrac{\bar{X}_A - \bar{X}_B}{\sqrt{\hat{S}(0)/n_A + \hat{S}(0)/n_B}}
</math>
where <math>\hat{S}(0)</math> is an estimate of the long-run variance (i.e., the spectral density at frequency zero), commonly estimated using [[Newey–West estimator|Newey-West estimators]] or batch means. Under the null hypothesis of convergence, the statistic <math>Z</math> follows an approximately standard normal distribution <math>Z \sim \mathcal{N}(0,1)</math>.

If <math>|Z| > 1.96</math>, the null hypothesis is rejected at the 5% significance level, suggesting that the chain has not yet reached stationarity.