A dimension reduction for extreme types of directed dependence

In recent years, a variety of novel measures of dependence have been introduced being capable of characterizing diverse types of directed dependence, hence diverse types of how a number of predictor variables X = ( X 1 , …, X p ), <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:math> p∈ N , may affect a response variable Y . This includes perfect dependence of Y on X and independence between X and Y , but also less well-known concepts such as zero-explainability, stochastic comparability and complete separation. Certain such measures offer a representation in terms of the Markov product ( Y , Y ′), with Y ′ being a conditionally independent copy of Y given X . This dimension reduction principle allows these measures to be estimated via the powerful nearest neighbor based estimation principle introduced in (Azadkia, M. and Chatterjee, S. (2021). A simple measure of conditional dependence. Ann. Stat. 49: 3070–3102). To achieve a deeper insight into the dimension reduction principle, this paper aims at translating the extreme variants of directed dependence, typically formulated in terms of the random vector ( X , Y ), into terms relating to its Markov product ( Y , Y ′).