Properties of the affine-invariant ensemble sampler's ‘stretch move’ in high dimensions

Abstract

We present theoretical and practical properties of the affine-invariant ensemble sampler Markov Chain Monte Carlo method. In high dimensions, the sampler's ‘stretch move’ has unusual and undesirable properties. We demonstrate this with an n-dimensional correlated Gaussian toy problem with a known mean and covariance structure, and a multivariate version of the Rosenbrock problem. Visual inspection of a trace plots suggests the burn-in period is short. Upon closer inspection, we discover the mean and the variance of the target distribution do not match the known values, and the chain takes a very long time to converge. This problem becomes severe as n increases beyond 50. We also applied different diagnostics adapted to be applicable to ensemble methods to determine any lack of convergence. The diagnostics include the Gelman–Rubin method, the Heidelberger–Welch test, the integrated autocorrelation and the acceptance rate. The trace plot of individual walkers appears to be useful as well. We therefore conclude that the stretch move should be used with caution in moderate to high dimensions. We also present some heuristic results explaining this behaviour.