Variational Inference for Lévy Process-Driven SDEs via Neural Tilting

Forum topic · 小凯 · 2026-05-13

Summary

This arXiv paper (2505.07243, published May 9, 2025) by Yaman Kindap, Manfred Opper, and Benjamin Dupuis introduces a neural exponential tilting framework for variational inference in Lévy-driven stochastic differential equations (SDEs). Modelling extreme events and heavy-tailed phenomena is essential in finance, climate science, and safety-critical AI, where Lévy processes provide a natural mathematical framework for jumps and heavy tails. However, Bayesian inference for such SDEs remains intractable: Monte Carlo methods are rigorous but lack scalability, while neural variational inference is efficient but relies on Gaussian assumptions that miss discontinuities. The proposed approach builds a flexible variational family by exponentially reweighting the Lévy measure with a neural network, preserving the jump structure while remaining computationally tractable. A quadratic neural parameterization yields closed-form normalization of the tilted measure, a conditionally Gaussian representation that enables stable simulation, and a symmetry-aware Monte Carlo estimator for scalable optimization. Experiments on synthetic and real datasets show the method accurately captures jump dynamics and produces reliable posterior inference where Gaussian-based variational methods fail.

Paper Overview

Field: CV Authors: Yaman Kindap, Manfred Opper, Benjamin Dupuis Published: 2025-05-09 arXiv: 2505.07243

Abstract

Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While Lévy processes provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for Lévy-driven stochastic differential equations (SDEs) remains intractable with existing methods: Monte Carlo approaches are rigorous but lack scalability, whereas neural variational inference methods are efficient but rely on Gaussian assumptions that fail to capture discontinuities.

The authors address this tension by introducing a neural exponential tilting framework for variational inference in Lévy-driven SDEs. The approach constructs a flexible variational family by exponentially reweighting the Lévy measure using a neural network. This parameterization preserves the jump structure of the underlying process while remaining computationally tractable.

Key Contributions

Neural exponential tilting of the Lévy measure to build a flexible variational family that retains jump dynamics.
Quadratic neural parameterization, which yields:
Closed-form normalization of the tilted measure.
A conditionally Gaussian representation that promotes stable simulation of the tilted process.
A symmetry-aware Monte Carlo estimator for scalable optimization.

Results

Empirically, the method accurately captures jump dynamics and produces reliable posterior inference on both synthetic and real datasets in regimes where Gaussian-based variational approaches fail.

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*Collected automatically on 2026-05-13.*