Kolmogorov-Smirnov Convergence Diagnostic


The Kolmogorov-Smirnov test is a nonparametric test of stationarity that has been applied as an MCMC diagnostic (Brooks et al, 2003), such as to the posterior samples from the LaplacesDemon function. The first and last halves of the chain are compared. This test assumes IID, which is violated in the presence of autocorrelation.

The KS.Diagnostic is a univariate diagnostic that is usually applied to each marginal posterior distribution. A multivariate form is not included. By chance alone due to multiple independent tests, 5% of the marginal posterior distributions should appear non-stationary when stationarity exists. Assessing multivariate convergence is difficult.





This is a vector of posterior samples for which a Kolmogorov-Smirnov test will be applied that compares the first and last halves for stationarity.


There are two main approaches to using the Kolmogorov-Smirnov test as an MCMC diagnostic. There is a version of the test that has been adapted to account for autocorrelation (and is not included here). Otherwise, the chain is thinned enough that autocorrelation is not present or is minimized, in which case the two-sample Kolmogorov-Smirnov test is applied. The CDFs of both samples are compared. The ks.test function in base R is used.

The advantage of the Kolmogorov-Smirnov test is that it is easier and faster to calculate. The disadvantages are that autocorrelation biases results, and the test is generally biased on the conservative side (indicating stationarity when it should not).


The KS.Diagnostic function returns a frequentist p-value, and stationarity is indicated when p > 0.05.


Statisticat, LLC. software@bayesian-inference.com


Brooks, S.P., Giudici, P., and Philippe, A. (2003). "Nonparametric Convergence Assessment for MCMC Model Selection". Journal of Computational and Graphical Statistics. 12(1), p. 1–22.

See Also

is.stationary, ks.test, and LaplacesDemon.



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