ABSTRACT
In this section we review the definitions and basic properties of conditional independence, which are crucial for Sufficient Dimension Reduction. Our development is in terms of conditional independence of σ $ \sigma $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math2_1.tif"/> -fields — rather than that of random variables — which is more general than can be found in most text books. Besides the generality, it is in fact easier to discuss conditional independence in terms of σ $ \sigma $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math2_2.tif"/> -fields than in terms of random variables. After developing the properties of conditional independence in the general setting, we then discuss the special cases where the relevant σ $ \sigma $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math2_3.tif"/> -fields are generated by random variables. For more information on conditional independence of σ $ \sigma $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math2_4.tif"/> -fields, see jorgensenSPSHYP1994 (jorgensenSPSHYP1994), page 460.
