# Adaptive Bayes exactly tracks information over intrinsic time

arXiv: https://arxiv.org/abs/2607.08789 · HTML: https://mlsanity.com/paper/2607.08789/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 73 references that are in the map.
- First posted on arXiv 26 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [A second order regret bound for NormalHedge](https://mlsanity.com/paper/2602.08151/) (2026)
- [Information from coincidences](https://mlsanity.com/paper/2606.25042/) (2026)
- [Sharp Finite-Time Iterated-Logarithm Martingale Concentration](https://mlsanity.com/paper/1405.2639/) (2014)
- [A Short Note on a Variant of the Squint Algorithm](https://mlsanity.com/paper/2603.03409/) (2026)

## Built on by (most cited first, 2 of 2)

- [The concentration game: Bayesian updating, regret, and information](https://mlsanity.com/paper/2608.18061/) (2026)
- [Exact information accounting for SGD methods](https://mlsanity.com/paper/2610.00446/) (2026)

## Abstract (arXiv)

Bayesian and multiplicative-weights updates reweight experts, models, or actions from sequential feedback. We show that the regret of any such update obeys an exact information-accounting identity. On each round, the learner's excess loss to any chosen comparator is the sum of an immediate cost for the uncertainty exposed by the round and a reduction in the information distance from the learner's current weights to the comparator. The cumulative cost defines a pathwise uncertainty clock, the intrinsic time of the realized sequence. Summing one-step balances yields two exact adaptive decompositions of cumulative regret, one for each natural way of composing the update across rounds. Because the decompositions are exact, favorable stochastic or low-noise regimes appear as self-bounding properties of the realized intrinsic time. The accounting also fixes a learning rate, inverse in the square root of intrinsic time. That schedule is competitive with adaptive baselines in selected online-learning settings. The same calculus covers Hedge, optimistic and side-information variants, continuous priors, boosting, online convex optimization, contextual bandits, and repeated games: the pathwise account is the same in every case.