Game-theoretic statistics and safe anytime-valid inference (2210.01948v2)

Abstract: Safe anytime-valid inference (SAVI) provides measures of statistical evidence and certainty -- e-processes for testing and confidence sequences for estimation -- that remain valid at all stopping times, accommodating continuous monitoring and analysis of accumulating data and optional stopping or continuation for any reason. These measures crucially rely on test martingales, which are nonnegative martingales starting at one. Since a test martingale is the wealth process of a player in a betting game, SAVI centrally employs game-theoretic intuition, language and mathematics. We summarize the SAVI goals and philosophy, and report recent advances in testing composite hypotheses and estimating functionals in nonparametric settings.

• The paper introduces SAVI by applying test martingales to sequential hypothesis testing, ensuring valid statistical inference under arbitrary stopping times.
• It integrates game theory with statistical methods to dynamically adapt sequential tests, overcoming the limitations of traditional p-value based approaches.
• The framework is practical for real-time applications such as multi-armed bandits and continuous data streams, promising enhanced reliability and robustness.

Game-Theoretic Statistics and Safe Anytime-Valid Inference

The paper explores the field of statistical inference through the lens of game theory, specifically focusing on Safe Anytime-Valid Inference (SAVI). SAVI is characterized by statistical measures that remain valid under continuous observation and arbitrary stopping times. This is pivotal for truly flexible data analysis, especially in modern, data-rich environments.

Core Ideas

The central construct facilitating SAVI is the test martingale, a concept originating from probability theory. A test martingale represents a gambler's betting strategy in a game against nature, whose cumulative wealth informs about the veracity of a statistical hypothesis. The real novelty here, however, is the application of this concept to statistical inference, moving away from traditional approaches like p-values, which have limitations in sequential and monitoring contexts.

Theoretical Foundations

SAVI provides statistical evidence via test martingales that are structurally likened to buying a ticket by a gambler, expecting a return if the null hypothesis is false. This feature of allowing anytime validity—permitting checks and decisions at arbitrary points in data collection—circumvents the historical pitfalls of premature statistical inference, allowing for more robust and reliable conclusions.

Game-theoretically, SAVI recontextualizes inference as a sequence of bets, where the null hypothesis configures the odds. Here, a Sequential Test Family can be adapted dynamically, thus inherently embodying the flexibility to abide by the principles of good statistical practices.

Methodological Advances

The paper synthesizes a framework, combining concepts of e-processes – sequences similarly to e-values but in a dynamic setting. SAVI methods, by harnessing test martingales, are able to offer stronger guarantees for type-I error under optional stopping. This is crucial for practical scenarios in machine learning and real-time data streams, where the traditional assumptions of fixed-sample size do not hold.

Moreover, the authors emphasize the concept of Reverse Information Projection (RIPr) and Universal Inference (UI) for handling composite hypotheses, enriching traditional statistical inference with mechanisms that offer robustness across varying and composite data generating processes.

Practical and Theoretical Implications

SAVI, with its game-theoretic underpinning, maps comfortably onto settings like multi-armed bandits, AI, and meta-analyses, where continuous adaptation and evidence accumulation are necessary. Their methodological rigor and adaptability present potentially transformative tools for fields where data is viewed as a continuous stream rather than sporadic fixes in time.

Future Directions

The landscape of statistical testing is significantly broadened by SAVI, offering numerous future exploration avenues. These include the development of domain-specific e-processes and extending foundational game-theoretic concepts within AI-based adaptative systems. Further exploration into the interplay between e-processes and classical statistical approaches like p-values could usher a comprehensive synthesis in statistical theory, benefiting various practical applications.

In conclusion, this paper's approach circumvents traditional statistical boundaries, proposing a robust and dynamic framework where the accuracy of statistical tests is no longer compromised by the limitations of traditional methods—creating a paradigm shift towards more reliable and effective analytics in a continuously evolving data science landscape.