On relative universality, regression operator, and conditional independence
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909580102991872 |
|---|---|
| author | Li, Bing Jones, Ben Artemiou, Andreas |
| author_facet | Li, Bing Jones, Ben Artemiou, Andreas |
| contents | The notion of relative universality with respect to a σ-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of this result in the existing literature. The existing definition of relative universality seems to be too strong for the proof to be valid. In this note we modify the definition of relative universality using the concept of ǫ-measurability, and rigorously establish the mentioned unbiasedness and Fisher consistency. The significance of this result is beyond its original context of sufficient dimension reduction, because relative universality allows us to use the regression operator to fully characterize conditional independence, a crucially important statistical relation that sits at the core of many areas and methodologies in statistics and machine learning, such as dimension reduction, graphical models, probability embedding, causal inference, and Bayesian estimation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On relative universality, regression operator, and conditional independence Li, Bing Jones, Ben Artemiou, Andreas Statistics Theory Methodology Machine Learning 62 G.3 The notion of relative universality with respect to a σ-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of this result in the existing literature. The existing definition of relative universality seems to be too strong for the proof to be valid. In this note we modify the definition of relative universality using the concept of ǫ-measurability, and rigorously establish the mentioned unbiasedness and Fisher consistency. The significance of this result is beyond its original context of sufficient dimension reduction, because relative universality allows us to use the regression operator to fully characterize conditional independence, a crucially important statistical relation that sits at the core of many areas and methodologies in statistics and machine learning, such as dimension reduction, graphical models, probability embedding, causal inference, and Bayesian estimation. |
| title | On relative universality, regression operator, and conditional independence |
| topic | Statistics Theory Methodology Machine Learning 62 G.3 |
| url | https://arxiv.org/abs/2504.11044 |