Variational Inference for Lévy Process-Driven SDEs via Neural Tilting
2026-05-11 • Machine Learning
Machine LearningArtificial IntelligenceComputer Vision and Pattern RecognitionRobotics
AI summaryⓘ
The authors introduce a new method to better understand and predict sudden big changes (jumps) in complex systems like finance or climate. Existing methods either do not handle these jumps well or are too slow for large problems. Their method uses neural networks to adjust mathematical models called Lévy processes, keeping the jump behavior while staying efficient to compute. They show this approach works better than previous ones on both made-up and real data when sharp changes are important.
Lévy processesstochastic differential equationsBayesian inferencevariational inferenceexponential tiltingheavy tailsjump processesneural networksMonte Carlo methodsposterior inference
Authors
Yaman Kindap, Manfred Opper, Benjamin Dupuis, Umut Simsekli, Tolga Birdal
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. We address this tension by introducing a neural exponential tilting framework for variational inference in Lévy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the Lévy measure using neural networks. This parametrization preserves the jump structure of the underlying process while remaining computationally tractable. To enable efficient inference, we develop a quadratic neural parametrization that yields closed-form normalization of the tilted measure, a conditional Gaussian representation for stable processes that facilitates simulation, and symmetry-aware Monte Carlo estimators for scalable optimization. Empirically, we demonstrate that the method accurately captures jump dynamics and yields reliable posterior inference in regimes where Gaussian-based variational approaches fail, on both synthetic and real-world datasets.