代数统计模型

Outliers and patterns of outliers in contingency tables with Algebraic Statistics

In this paper we provide a definition of pattern of outliers in contingency tables within a model-based framework. In particular, we make use of log-linear models and exact goodness-of-fit tests to specify the notions of outlier and pattern of outliers. The language and some techniques from Algebraic Statistics are essential tools to make the definition clear and easily applicable. Some numerical examples show how to use our definitions.

Privacy and Statistical Risk: Formalisms and Minimax Bounds

We explore and compare a variety of definitions for privacy and disclosure limitation in statistical estimation and data analysis, including (approximate) differential privacy, testing-based definitions of privacy, and posterior guarantees on disclosure risk. We give equivalence results between the definitions, shedding light on the relationships between different formalisms for privacy. We also take an inferential perspective, where---building off of these definitions---we provide minimax risk bounds for several estimation problems, including mean estimation, estimation of the support of a di…

部分正则化普通最小二乘插值器的代数与统计性质

Algebraic and Statistical Properties of the Partially Regularized Ordinary Least Squares Interpolator

本文研究部分正则化OLS插值器的代数与统计性质，扩展Cochran公式与留一残差公式，提出同方差方差估计量，并通过模拟验证其有限样本性能。

A definition of qualitative robustness for general point estimators, and examples

A definition of qualitative robustness for point estimators in general statistical models is proposed. Some criteria for robustness are established and applied to estimators in parametric, semiparametric, and nonparametric models. In specific nonparametric models, the proposed definition boils down to Hampel robustness. It is also explained how plug-in estimators in certain nonparametric models can be reasonably classified w.r.t. their degrees of robustness.

Derivation of Born Rule from Algebraic and Statistical Axioms

In the present paper we propose a new axiomatic system of algebraic and statistical axioms as working hypotheses, from which Born rule can be seen to emerge. In this process the concept of sectors defined as quasi-equivalence classes of factor states plays a crucial role.

统计力学与机器学习理论等价框架中的代数研究：回顾与展望

Review and Prospect of Algebraic Research in Equivalent Framework between Statistical Mechanics and Machine Learning Theory

本文面向理论物理学家回顾并展望统计力学与机器学习理论等价框架中的代数研究，强调代数方法处理奇异随机哈密顿量以推导自由能与泛化误差，并介绍基于代数学习理论构建AI对齐理论基础的最新进展。