# On conditional versus marginal bias in multi-armed bandits

arXiv: https://arxiv.org/abs/2002.08422 · HTML: https://mlsanity.com/paper/2002.08422/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Feb 2020.

## Built on by (most cited first, 9 of 9)

- [Off-Policy Evaluation via Adaptive Weighting with Data from Contextual Bandits](https://mlsanity.com/paper/2106.02029/) (2021)
- [A Closer Look at the Worst-case Behavior of Multi-armed Bandit Algorithms](https://mlsanity.com/paper/2106.02126/) (2021)
- [Off-policy estimation of linear functionals: Non-asymptotic theory for semi-parametric efficiency](https://mlsanity.com/paper/2209.13075/) (2022)
- [Learning from an Exploring Demonstrator: Optimal Reward Estimation for Bandits](https://mlsanity.com/paper/2106.14866/) (2021)
- [A unified framework for bandit multiple testing](https://mlsanity.com/paper/2107.07322/) (2021)
- [Challenges in Statistical Analysis of Data Collected by a Bandit Algorithm: An Empirical Exploration in Applications to Adaptively Randomized Experiments](https://mlsanity.com/paper/2103.12198/) (2021)
- [Causally Abstracted Multi-armed Bandits](https://mlsanity.com/paper/2404.17493/) (2024)
- [Entropy Regularization for Population Estimation](https://mlsanity.com/paper/2208.11747/) (2022)
- [Reciprocal Learning](https://mlsanity.com/paper/2408.06257/) (2024)

## Abstract (arXiv)

The bias of the sample means of the arms in multi-armed bandits is an important issue in adaptive data analysis that has recently received considerable attention in the literature. Existing results relate in precise ways the sign and magnitude of the bias to various sources of data adaptivity, but do not apply to the conditional inference setting in which the sample means are computed only if some specific conditions are satisfied. In this paper, we characterize the sign of the conditional bias of monotone functions of the rewards, including the sample mean. Our results hold for arbitrary conditioning events and leverage natural monotonicity properties of the data collection policy. We further demonstrate, through several examples from sequential testing and best arm identification, that the sign of the conditional and marginal bias of the sample mean of an arm can be different, depending on the conditioning event. Our analysis offers new and interesting perspectives on the subtleties of assessing the bias in data adaptive settings.