Differential Geometry, Profile Likelihood, $L$-Sufficiency and Composite Transformation Models

研究问题: 如何将完整参数空间的信息几何结构转移到目标参数空间，并理解这些结构与剖面、修正剖面及边际似然的联系？
方法与思路: 作者考察期望与观测信息度量及相应几何联络，在特定条件下作规范转移，再联系伪似然导数和 L 充分统计量。
材料陈述的贡献: 摘要称，在复合变换模型中，修正剖面似然与某个边际似然积分表示的 Laplace 近似十分接近，并建立相关几何联系。
阅读关注点: 应核对几何结构转移的条件、L 充分统计量存在要求，以及边际似然积分表示和两种近似接近的具体含义与精度。
阅读边界: 仅据版本未注明的 OpenAlex 索引摘要解读。摘要未列出转移条件或近似误差阶数；不能将复合变换模型中的关系推广到任意模型和任意剖面似然。

GPT-6.1 Sol · 高 · 2026-10-05

原文摘要

Let $\Omega$ denote the parameter space of a statistical model and let $\mathscr{K}$ be the domain of variation of the parameter of interest. Various differential-geometric structures on $\Omega$ are considered, including the expected information metric and the $\alpha$ -connections studied by Chentsov and Amari, as well as the observed information metric and the observed $\alpha$ -connections introduced by Barndorff-Nielsen. Under certain conditions these geometric objects on $\Omega$ can be transferred in a canonical purely differential-geometric way to $\mathscr{K}$ . The transferred objects are related to structures on $\mathscr{K}$ obtained from derivatives of pseudolikelihood functions such as the profile likelihood, the modified profile likelihood and the marginal likelihood based on an $L$ -sufficient statistic (cf. Remon) when such a statistic exists. For composite transformation models it is shown that the modified profile likelihood is very close to the Laplace approximation to a certain integral representation of the marginal likelihood.Let $\Omega$ denote the parameter space of a statistical model and let $\mathscr{K}$ be the domain of variation of the parameter of interest. Various differential-geometric structures on $\Omega$ are considered, including the expected information metric and the $\alpha$-connections studied by Chentsov and Amari, as well as the observed information metric and the observed $\alpha$-connections introduced by Barndorff-Nielsen. Under certain conditions these geometric objects on $\Omega$ can be transferred in a canonical purely differential-geometric way to $\mathscr{K}$. The transferred objects are related to structures on $\mathscr{K}$ obtained from derivatives of pseudolikelihood functions such as the profile likelihood, the modified profile likelihood and the marginal likelihood based on an $L$-sufficient statistic (cf. Remon) when such a statistic exists. For composite transformation models it is shown that the modified profile likelihood is very close to the Laplace approximation to a certain integral representation of the marginal likelihood.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）

笔记与分组

分类依据与纠错

Differential Geometry, Profile Likelihood, LLL-Sufficiency and Composite Transformation ModelsDifferential Geometry, Profile Likelihood, $L$-Sufficiency and Composite Transformation Models

代码、数据与补充材料

微分几何、剖面似然、L 充分性与复合变换模型

原文摘要

Differential Geometry, Profile Likelihood, $L$ -Sufficiency and Composite Transformation ModelsDifferential Geometry, Profile Likelihood, $L$-Sufficiency and Composite Transformation Models