Mechanistic explanation for Barker proposal’s improved performance via reflection

Ascertain whether the improved mode-finding at small step-sizes observed for the Barker proposal on the univariate log-quartic target π(x) ∝ exp(−x^4) is indeed caused by the reflection mechanism that yields better‑informed proposal jumps and fewer rejections.

Background

The Barker proposal constructs moves by pre-proposing Gaussian noise and then deciding, per coordinate, whether to add or reflect this noise using a logistic function of the local gradient, before a Metropolis correction.

In experiments on the log‑quartic target, the authors observed faster convergence to the mode at small step-sizes compared to other methods and conjecture that the reflection step is responsible for this improvement. Verifying this mechanistic explanation would help guide the design and tuning of Barker-type proposals.