[Paper Review] Properties of the Affine Invariant Ensemble Sampler in high dimensions

David Huijser, Jesse Goodman+1 more|arXiv (Cornell University)|Sep 7, 2015

Markov Chains and Monte Carlo Methods15 references20 citations

TL;DR

This paper investigates the performance of the affine-invariant ensemble sampler's 'stretch move' in high-dimensional problems, revealing that it exhibits severe convergence issues due to poor mixing and biased estimates. Despite appearing to converge quickly via trace plots, the sampler fails to accurately recover the true mean and variance in high dimensions (n > 50), especially in correlated Gaussian and Rosenbrock targets, indicating the stretch move should not be relied upon in isolation for moderate to high-dimensional inference.

ABSTRACT

We present theoretical and practical properties of the affine-invariant ensemble sampler Markov chain Monte Carlo method. In high dimensions the affine-invariant ensemble sampler shows unusual and undesirable properties. We demonstrate this with an $n$-dimensional correlated Gaussian toy problem with a known mean and covariance structure, and analyse the burn-in period. The burn-in period seems to be short, however upon closer inspection we discover the mean and the variance of the target distribution do not match the expected, known values. This problem becomes greater as $n$ increases. We therefore conclude that the affine-invariant ensemble sampler should be used with caution in high dimensional problems. We also present some theoretical results explaining this behaviour.

Motivation & Objective

• To investigate the behavior of the affine-invariant ensemble sampler's stretch move in high-dimensional target distributions.
• To identify why the sampler appears to converge quickly but fails to produce accurate posterior estimates in high dimensions.
• To evaluate diagnostic tools for detecting slow convergence in the absence of known true distributions.
• To provide theoretical and empirical evidence that the stretch move becomes ineffective as dimensionality increases beyond 50.
• To caution practitioners against relying on the AIES stretch move in high-dimensional settings without rigorous convergence checks.

Proposed method

• The study uses a high-dimensional correlated Gaussian target with known mean and covariance to test convergence properties.
• A multivariate Rosenbrock distribution is used as a second test case to assess performance on non-Gaussian, strongly correlated targets.
• The stretch move is implemented using the emcee Python library, with walkers updated via affine-invariant proposals based on uniform selection of complementary walkers.
• The acceptance probability incorporates a $ Z^{n-1} $ correction factor to account for the one-dimensional proposal distribution in n-dimensional space.
• Diagnostic tools include trace plots of selected walkers, histograms of accepted $ Z $ values, and the adapted Gelman-Rubin $ R-1 $ statistic.
• Theoretical analysis explores how the $ Z^{n-1} $ factor distorts proposal efficiency in high dimensions, leading to poor mixing and slow convergence.

Experimental results

Research questions

• RQ1 Why does the AIES stretch move appear to converge quickly in high dimensions despite poor mixing and inaccurate estimates?
• RQ2 How do the acceptance rates and $ Z $-value distributions change with increasing dimensionality in the AIES algorithm?
• RQ3 To what extent do trace plots mislead users into thinking convergence has occurred when it has not?
• RQ4 What diagnostic indicators can reliably detect slow convergence in the AIES when the true target distribution is unknown?
• RQ5 How does the performance of the stretch move degrade in strongly correlated, high-dimensional distributions like the multivariate Rosenbrock function?

Key findings

• The AIES stretch move fails to converge to the correct mean and variance in high-dimensional correlated Gaussian targets, even when trace plots suggest otherwise.
• For $ n > 50 $, the sampler exhibits extremely slow mixing, with individual walkers confined to narrow regions of parameter space despite apparent stability in trace plots.
• The distribution of accepted $ Z $ values becomes bimodal, clustering around $ Z=1 $, indicating that proposals are rarely accepted far from the current position.
• The adapted Gelman-Rubin diagnostic shows poor convergence, with $ R-1 $ values remaining high even after long runs, signaling lack of effective exploration.
• Trace plots of individual walkers are misleading: while they appear well-mixed at $ n=10 $, they show restricted movement and low relative ordering changes at $ n=50 $ and $ n=100 $.
• Theoretical analysis confirms that the $ Z^{n-1} $ factor in the acceptance ratio suppresses proposals far from the current walker, especially in high dimensions, leading to poor exploration efficiency.