Channelling Multimodality Through a Unimodalizing Transport: Warp-U Sampler and Stochastic Bridge Sampling Estimator (2401.00667v2)

Abstract: Monte Carlo integration is a powerful tool for scientific and statistical computation, but faces significant challenges when the integrand is a multi-modal distribution, even when the mode locations are known. This work introduces novel Monte Carlo sampling and integration estimation strategies for the multi-modal context by leveraging a generalized version of the stochastic Warp-U transformation Wang et al. [2022]. We propose two flexible classes of Warp-U transformations, one based on a general location-scale-skew mixture model and a second using neural ordinary differential equations. We develop an efficient sampling strategy called Warp-U sampling, which applies a Warp-U transformation to map a multi-modal density into a uni-modal one, then inverts the transformation with injected stochasticity. In high dimensions, our approach relies on information about the mode locations, but requires minimal tuning and demonstrates better mixing properties than conventional methods with identical mode information. To improve normalizing constant estimation once samples are obtained, we propose a stochastic Warp-U bridge sampling estimator, which we demonstrate has higher asymptotic precision per CPU second compared to the original approach proposed by Wang et al. [2022]. We also establish the ergodicity of our sampling algorithm. The effectiveness and current limitations of our methods are illustrated through simulation studies and an application to exoplanet detection.