Information-Theoretic Martingales: Unifying Sequential Inference and Entropy Production

Recent work by Nadal-Rosa & Manzano (2025) leverages martingale theory for thermodynamic inference in molecular motors, using fluctuation theorems at stopping times. Meanwhile, Ramdas et al. (2020) show the centrality of nonnegative martingales for anytime-valid sequential inference. This idea brings these strands together: systematically develop an information-theoretic martingale framework where quantities like entropy, mutual information, or divergence are tracked as martingale processes, with stopping times interpreted as information thresholds or decision boundaries. This could yield new sequential tests or confidence sequences with explicit information-theoretic guarantees, and new fluctuation theorems in stochastic thermodynamics. Such an approach would not only bridge probability and information theory, but also suggest novel algorithms for change detection, online learning, or adaptive experimental design.

References:

1. Admissible anytime-valid sequential inference must rely on nonnegative martingales.. Aaditya Ramdas, J. Ruf, Martin Larsson, Wouter M. Koolen (2020).
2. Thermodynamic inference in molecular motors: A martingale approach. Adri'an Nadal-Rosa, Gonzalo Manzano (2025). Physical Review E.