Impact of inhomogeneous component scaling on optimal RWM acceptance rate in low dimensions

Determine, for random-walk Metropolis samplers targeting multimodal distributions with inhomogeneous component-wise scaling (e.g., targets of the form π(x) = ∏_{i=1}^d C_i f(C_i x_i) with independent scaling factors C_i), the extent to which such scaling reduces the ESJD-maximizing acceptance rate below approximately 0.234 in low-dimensional settings, and identify the mechanisms responsible for this reduction.

Background

The paper reports empirical findings that for multimodal targets with inhomogeneous component-wise scaling, the acceptance rate maximizing ESJD in random-walk Metropolis can be substantially below 0.234 in smaller dimensions, while tending toward 0.234 as dimension increases. This contrasts with homogeneous or i.i.d. product targets where 0.234 remains robust even at moderate dimensions.

The authors hypothesize that the deviation in low dimensions may be due to anisotropy introduced by the scaling factors and limited averaging effects, but explicitly state that the extent and reasons for this phenomenon remain unresolved. Clarifying this would sharpen practical guidance for tuning RWM in problems with heterogeneously scaled components.