As Basu (1977) writes, "Eliminating nuisance parameters from a model is universally recognized as a major problem of statistics," but after more than 50 years since Basu wrote these words, the two mainstream schools of thought in statistics have yet to solve the problem. Fortunately, the two mainstream frameworks aren't the only options. This series of papers rigorously develops a new and very general inferential model (IM) framework for imprecise-probabilistic statistical inference that is provably valid and efficient, while simultaneously accommodating incomplete or partial prior information about the relevant unknowns when it's available. The present paper, Part III in the series, tackles the marginal inference problem. Part II showed that, for parametric models, the likelihood function naturally plays a central role and, here, when nuisance parameters are present, the same principles suggest that the profile likelihood is the key player. When the likelihood factors nicely, so that the interest and nuisance parameters are perfectly separated, the valid and efficient profile-based marginal IM solution is immediate. But even when the likelihood doesn't factor nicely, the same profile-based solution remains valid and leads to efficiency gains. This is demonstrated in several examples, including the famous Behrens--Fisher and gamma mean problems, where I claim the proposed IM solution is the best solution available. Remarkably, the same profiling-based construction offers validity guarantees in the prediction and non-parametric inference problems. Finally, I show how a broader view of this new IM construction can handle non-parametric inference on risk minimizers and makes a connection between non-parametric IMs and conformal prediction.

翻译：正如 Basu（1977年）所言："从模型中消除冗余参数被普遍认为是统计学的一个主要问题"，但在这句话发表50多年后，统计学两大主流学派仍未解决该问题。幸运的是，这两大主流框架并非唯一选择。本系列论文严谨地发展了一种全新且极为通用的推断模型（IM）框架，用于可证明有效且高效的精概率统计推断，同时能够灵活处理相关未知量存在的不完整或部分先验信息（如有可用信息）。本篇作为系列的第三部分，聚焦于边缘推断问题。第二部分已表明，对于参数模型而言，似然函数自然扮演核心角色；而在存在冗余参数的情况下，相同原理表明剖面似然是关键要素。当似然函数具有良好分解性（即关注参数与冗余参数完全分离）时，基于剖面的有效且高效边缘IM解即可直接获得。即便似然函数无法良好分解，相同的基于剖面的解仍保持有效性，并可带来效率提升。这一结论通过多个案例得到验证，包括著名的贝伦斯-费舍尔问题和伽马均值问题——笔者主张所提出的IM解是现有最优解。值得关注的是，同一基于剖面的构建方法在预测和非参数推断问题中也能提供有效性保证。最后，本文展示了这种新型IM构建的广义视角如何用于处理风险最小化器的非参数推断，并在非参数IM与共形预测之间建立了联系。