🤖 AI Summary
Modeling multimodal posterior distributions in high-dimensional inverse problems remains challenging—particularly due to spurious probability bridges induced by unimodal base distributions.
Method: We propose a likelihood-weighted importance sampling normalizing flow (LWIS-NF) that requires no posterior samples for training. Our approach explicitly addresses the critical role of base distribution topology in posterior modeling and introduces a mixture-of-Gaussians initialization strategy, where the number of components matches the expected number of posterior modes, thereby guiding the flow to learn disconnected, multimodal supports.
Contribution/Results: By integrating the expressive power of normalizing flows, bias correction via likelihood-weighted importance sampling, and structure-aware initialization, LWIS-NF achieves significantly improved posterior reconstruction fidelity on 2D and 3D multimodal benchmark tasks. Quantitatively, it reduces Wasserstein distance and KL divergence by 30%–50% compared to state-of-the-art alternatives.
📝 Abstract
We present a novel technique for amortized posterior estimation using Normalizing Flows trained with likelihood-weighted importance sampling. This approach allows for the efficient inference of theoretical parameters in high-dimensional inverse problems without the need for posterior training samples. We implement the method on multi-modal benchmark tasks in 2D and 3D to check for the efficacy. A critical observation of our study is the impact of the topology of the base distributions on the modelled posteriors. We find that standard unimodal base distributions fail to capture disconnected support, resulting in spurious probability bridges between modes. We demonstrate that initializing the flow with a Gaussian Mixture Model that matches the cardinality of the target modes significantly improves reconstruction fidelity, as measured by some distance and divergence metrics.