# Sharp Characterization of Bias in Post-Bandit Inference

arXiv: https://arxiv.org/abs/2608.01069 · HTML: https://mlsanity.com/paper/2608.01069/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 41 references that are in the map.
- First posted on arXiv 2 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [On the bias, risk and consistency of sample means in multi-armed bandits](https://mlsanity.com/paper/1902.00746/) (2019)
- [UCB algorithms for multi-armed bandits: Precise regret and adaptive inference](https://mlsanity.com/paper/2412.06126/) (2024)
- [The Fragility of Optimized Bandit Algorithms](https://mlsanity.com/paper/2109.13595/) (2021)
- [Inference with the Upper Confidence Bound Algorithm](https://mlsanity.com/paper/2408.04595/) (2024)
- [On Instability of Minimax Optimal Optimism-Based Bandit Algorithms](https://mlsanity.com/paper/2511.18750/) (2025)
- [Bandit Allocational Instability](https://mlsanity.com/paper/2602.07472/) (2026)

## Abstract (arXiv)

Bandit algorithms generate data for downstream inference, but adaptive sampling biases post-bandit sample means. We analyze this bias for stable index algorithms, including UCB1 and its generalizations, and derive sharp leading-order expressions for the sample-mean bias and expected Z-statistic, in bandit experiments of fixed horizon T. Our characterization reveals the algorithmic origin of bias through a key index-function-dependent quantity, which we term effective exploration rate. For example, under UCB1, the effective exploration rate is of order √log T, and the standardized bias of any arm (that is not uniquely optimal) decays at the extremely slow rate 1/√log T. We also show how the choice of the index function affects both regret and bias, which reveals a regret-bias trade-off: more exploratory algorithm reduces bias but increases regret. We further show how bias most severely distorts confidence intervals and hypothesis tests when the tested arm is one of the tied-optimal arms. Our sharp characterization for bias uses a novel empirical fluid approximation of the algorithm's sampling dynamics, which may be of independent interest.