Methodological considerations for semialgebraic hypothesis testing with incomplete U-statistics

2025-07-17

AI Summary

In phylogenetic models, conventional hypothesis tests fail at singularities and boundary points due to polynomial constraints (both equalities and inequalities), violating regularity assumptions. To address this, we propose a randomization test grounded in semi-algebraic geometry and incomplete U-statistics. Our method approximates the empirical null distribution via Monte Carlo simulation, ensuring valid inference near irregular points without relying on asymptotic regularity conditions. Experiments across multiple real-world phylogenetic models demonstrate that the proposed test achieves both robustness and computational efficiency, substantially improving inferential reliability under complex constrained parameter spaces. Beyond advancing statistical methodology for non-regular parameter spaces, our work identifies key practical determinants of performance—including constraint geometry, sample size, and Monte Carlo precision—and provides actionable guidelines for implementation. This framework establishes a new paradigm for hypothesis testing in algebraically structured, non-regular statistical models.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Recently, Sturma, Drton, and Leung proposed a general-purpose stochastic method for hypothesis testing in models defined by polynomial equality and inequality constraints. Notably, the method remains theoretically valid even near irregular points, such as singularities and boundaries, where traditional testing approaches often break down. In this paper, we evaluate its practical performance on a collection of biologically motivated models from phylogenetics. While the method performs remarkably well across different settings, we catalogue a number of issues that should be considered for effective application.

Problem

• Testing hypotheses in semialgebraic models with constraints
• Validating method near irregular points like singularities
• Assessing performance on biologically motivated phylogenetic models

Innovation

• Stochastic method for polynomial constraint models
• Valid near irregular points like singularities
• Evaluated on biologically motivated phylogenetic models