Betting on Moments: Legendre Jumper Martingales for Online Exchangeability Testing

A fundamental assumption in statistics and machine learning is that ``the future looks like the past,'' formalized as exchangeability: the joint data distribution is order-invariant. In practice, this assumption is often violated due to distribution shifts over time. Early detection of exchangeability violations is crucial to prevent performance degradation and enable timely interventions like model retraining. Conformal test martingales offer a flexible, distribution-free framework for sequential exchangeability testing with guaranteed false-alarm rate control by betting against the uniformity of conformal p-values. While alternatives such as plug-in martingales and mixture-based strategies exist, computationally efficient baselines like the Simple Jumper are limited to detecting mean location shifts. We propose a family of conformal test martingales based on shifted Legendre polynomials that extend the Simple Jumper to higher-order moments. The Simple Legendre Jumper replaces linear betting functions with polynomials of arbitrary degree, enabling rapid detection of variance, skewness, and other higher-order deviations. The Product Legendre Jumper combines multiple polynomial degrees into a single betting function but suffers from exponential state-space growth, termed the jumping tax. To resolve this, we introduce the Variational Legendre Jumper, which employs a mean-field approximation to reduce complexity to constant time per step with minimal power loss, providing an expressive, scalable framework for real-time distribution shift monitoring.