Information on Trajectories: Martingales and Random Times (arXiv 2608.20337)

Forum topic · 小凯 · 2026-08-24

Summary

A new paper by Akshay Balsubramani (arXiv 2608.20337) models information flow on the path space of trajectories of a nonnegative martingale, deriving exact variational identities that hold even at arbitrary random times. This framework recovers widely used classical concentration inequalities, from Ville's inequality to PAC-Bayes bounds, and quantifies exactly what information each inequality discards. The controlled tail is itself a relative entropy that decomposes via the chain rule into per-step conditional divergences. The discarded slack terms take precise forms in three geometries: the Gibbs tilt of Azuma-Hoeffding and PAC-Bayes bounds, the crossing itself in Ville's inequality and merging tests, and control certificates for L^p maximal inequalities, whose optional-stopping deficit decomposes stepwise into Bregman divergences of the running maximum. On path-time space, the identity gains a pricing expectation factor: any random time incurs an e-process "peeking penalty." The partition function admits an interpretation as a coalescent process, the prefix-sharing probability of independent copies, and geometric mixtures of test martingales yield merging benefits for multi-model safe testing.

Paper Overview

Research area: ML Author: Akshay Balsubramani Published: 2026-08-22 arXiv: 2608.20337

Abstract

Accounting for information flow on the path space of trajectories of a nonnegative martingale yields exact variational identities for it, even at arbitrary random times. This recovers the widely used classical concentration inequalities, from Ville to PAC-Bayes, and measures what each one discards.

Key points (from the extended summary)

The framework models information flow on path spaces of nonnegative martingales and yields exact variational identities, valid even at arbitrary (random) stopping times.
It recovers widely used classical concentration inequalities, from Ville's inequality to PAC-Bayes bounds, and quantifies the information discarded by each.
The controlled tail is itself a relative entropy, decomposing by the chain rule into per-step conditional divergences.
The discarded relaxation terms have precise forms in three geometries:
The Gibbs tilt underlying Azuma-Hoeffding and PAC-Bayes bounds.
The crossing itself, in Ville's inequality and in merging tests.
Control certificates for L^p maximal inequalities; their optional-stopping deficit decomposes stepwise into Bregman divergences of the running maximum.
On path-time space, the same identity gains a factor pricing expectations: any random time carries an e-process "peeking penalty."
The partition function can be read as a coalescent process — the prefix-sharing probability of independent copies — and geometric mixtures of test martingales yield merging benefits for multi-model safe testing.

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