Estimating means of bounded random variables by betting (2010.09686v7)

Abstract: This paper derives confidence intervals (CI) and time-uniform confidence sequences (CS) for the classical problem of estimating an unknown mean from bounded observations. We present a general approach for deriving concentration bounds, that can be seen as a generalization and improvement of the celebrated Chernoff method. At its heart, it is based on a class of composite nonnegative martingales, with strong connections to testing by betting and the method of mixtures. We show how to extend these ideas to sampling without replacement, another heavily studied problem. In all cases, our bounds are adaptive to the unknown variance, and empirically vastly outperform existing approaches based on Hoeffding or empirical Bernstein inequalities and their recent supermartingale generalizations. In short, we establish a new state-of-the-art for four fundamental problems: CSs and CIs for bounded means, when sampling with and without replacement.

• The paper introduces a betting-inspired framework that employs martingale methods to construct confidence intervals for bounded random variables.
• It presents variance-adaptive bounds and techniques for without-replacement sampling, outperforming traditional Hoeffding and empirical Bernstein methods.
• Empirical results demonstrate that the new strategies yield tighter confidence sequences and more efficient sequential inference.

Overview of "Estimating means of bounded random variables by betting"

The paper by Waudby-Smith and Ramdas from Carnegie Mellon University presents a novel approach to constructing confidence intervals (CIs) and confidence sequences (CSs) for estimating the mean of bounded random variables. The authors employ betting strategies based on martingales to derive these intervals, demonstrating theoretical advancements and empirical superiority over classical methods such as Hoeffding's and empirical Bernstein inequalities.

Core Contributions

The paper offers several key contributions that push the boundaries of existing methodologies in statistical inference:

1. Generalization of Martingale Methods: The authors leverage a class of composite nonnegative martingales to construct concentration bounds. This approach builds upon and extends the traditional Chernoff method, providing a robust framework to derive CIs and CSs that are valid for stopping times.
2. Variance-Adaptation: The proposed bounds are adaptive to the unknown variance of the data, allowing for more efficient inference compared to fixed bounds like Hoeffding's, especially in low-variance settings.
3. Betting Interpretation: The paper presents a reinterpretation of statistical testing and estimation as a betting problem. This intuitive framework allows for flexible betting strategies that can be tuned to maximize the expected growth rate of capital, thereby yielding tighter confidence bounds.
4. Without Replacement Sampling: The authors extend their framework to handle sampling without replacement, a situation traditionally more challenging due to dependencies introduced by the sampling process. Their methods show significant improvements over classical approaches in this domain.

Theoretical and Empirical Results

The paper rigorously develops several different betting-based strategies, including predictable plug-ins and hedged betting strategies, to construct their martingale-based bounds. Notably, the authors present several scenarios where these new methods significantly outperform classical results:

• Empirical Bernstein Confidence Sequences: These are derived using predictable plug-ins that adapt to the empirical variance, providing efficient closed-form solutions that match or exceed the performance of mixture methods, with substantial computational advantages.
• Hedged Capital Process: This novel strategy involves simultaneous bets on multiple hypotheses (means) and selecting the maximum of the accumulated capital processes. It leads to both practical and theoretical improvements, offering CSs that adapt well to both variance and data asymmetry.

Empirically, the proposed methods demonstrate strong performance across various simulations, outperforming existing techniques like the Hoeffding-based methods, particularly in low-variance settings. The empirical tests cover both i.i.d observations and scenarios involving sampling without replacement, demonstrating the robustness and versatility of the new approaches.

Implications and Future Directions

The advancements presented in this paper have several important implications:

• Practical Applications: The derivation of more efficient and tighter confidence intervals has immediate applications in fields like sequential analysis and online learning, where quick decision-making is crucial.
• Theoretical Insights: The betting framework provides new insights into the nature of statistical inference itself, challenging conventional paradigms and offering a fresh perspective on traditional problems.
• Future Research: The techniques can be extended to other distributional settings, beyond bounded random variables, presenting exciting opportunities for future research. Moreover, the relationship between these methods and existing ones in information theory and online learning suggests potential cross-disciplinary advancements.

In summary, this work provides a solid advancement over classical statistical methods, offering both conceptual insights and practical tools for modern statistical challenges. The infusion of game-theoretic probability with statistical inference reimagines the landscape of nonparametric estimation, possibly heralding new directions in research and application.