Confidence Estimation via Sequential Likelihood Mixing (2502.14689v1)

Abstract: We present a universal framework for constructing confidence sets based on sequential likelihood mixing. Building upon classical results from sequential analysis, we provide a unifying perspective on several recent lines of work, and establish fundamental connections between sequential mixing, Bayesian inference and regret inequalities from online estimation. The framework applies to any realizable family of likelihood functions and allows for non-i.i.d. data and anytime validity. Moreover, the framework seamlessly integrates standard approximate inference techniques, such as variational inference and sampling-based methods, and extends to misspecified model classes, while preserving provable coverage guarantees. We illustrate the power of the framework by deriving tighter confidence sequences for classical settings, including sequential linear regression and sparse estimation, with simplified proofs.

• The paper introduces a sequential likelihood mixing framework that constructs (1-δ) confidence sequences by unifying Bayesian inference and frequentist principles.
• It employs sequential likelihood ratios and smooth approximations from variational inference to offer tractable and robust confidence estimation without relying on i.i.d. assumptions.
• Its framework maintains coverage guarantees under model misspecification and is applicable to diverse scenarios like sequential regression and online decision-making.

Confidence Estimation via Sequential Likelihood Mixing

The paper "Confidence Estimation via Sequential Likelihood Mixing" by Kirschner et al. introduces a comprehensive framework for creating confidence sets through a process referred to as sequential likelihood mixing. This framework assimilates classical principles from sequential analysis, establishing a profound interconnection between Bayesian inference, sequential mixing, and regret inequalities prevalent in online estimation domains. The methodology is applicable to any realizable family of likelihood functions and can accommodate non-i.i.d. data, assuring anytime validity. Additionally, the approach is versatile, integrating smooth approximations commonly used in inference procedures like variational inference and sampling methods, and maintaining coverage guarantees even with model misspecification.

Framework Overview

The paper outlines a universal framework for constructing confidence sequences, a tool crucial for model uncertainty quantification, pivotal in domains such as medical diagnosis, autonomous driving, and reinforcement learning. The ability to compute confidence sets that are valid at any interim point and do not rely on asymptotic properties is a well-recognized challenge in complex, data-dependent settings.

Technical Contributions

This work revisits and extends the use of classical likelihood ratio martingales, enhanced through the concept of likelihood mixing, to yield a unified approach capable of constructing valid $(1-\delta)$ confidence sequences. The paper emphasizes on several critical points:

• Sequential Likelihood Ratios: By evaluating sequential likelihood ratios, the paper employs Ville's inequality to construct confidence sets without necessitating i.i.d. assumptions or specific parametric requisites on the model, bridging the gap between traditional frequentist and Bayesian perspectives.
• Integration with Approximate Inference: Techniques like variational inference are seamlessly integrated into the framework, enabling tractable computation of confidence coefficients. This eradicates reliance on precise model-likelihood knowledge, a significant advantage for practical applications.
• Sequential Mixing: By defining data-dependent confidence coefficients through the use of sequential mixing distributions over likelihood ratio martingales, which assimilates Bayesian updating and frequentist confidence estimation approaches, the authors propose a universally applicable method that confers robust performance over a broad scope of scenarios.

Numerical and Theoretical Implications

This approach facilitates deriving tighter confidence sequences pertinent to typical problem settings such as sequential linear regression. By exploring applications, including domains necessitating sparse estimation, this methodology provides simplified yet theoretically grounded results, while fostering connections between Bayesian posterior approximations and core frequentist principles.

Future Trajectories and Applications

The theoretical backdrop paved by this research envisions multiple avenues for further exploration:

• Extension to Misspecified Models: Relaxing realizability assumptions to misspecified models, or those with approximate likelihoods, remains a promising area for developing robust statistical tools in real-world settings.
• Enhanced Online Estimation: Leveraging regret bounds from online convex optimization contributes toward more refined, practically implementable algorithms for sequential decision-making under uncertainty.
• Broader Application Integration: Applying these confidence sequences into broader machine learning tasks such as active learning, and exploring non-parametric or complex function settings is poised to significantly impact model-safe deployment strategies.

In summary, by presenting a framework that harmonizes sequential analysis with Bayesian and frequentist methodologies through likelihood mixing, the authors mark a significant leap in establishing reliable confidence estimation protocols. These sequences not only hold the potential to enhance existing applications but also pivot towards incorporating more structured, data-adaptive inference techniques into the core machine learning toolbox.