A dimension reduction for extreme types of directed dependence

Abstract: 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 $\mathbf{X} = (X_1, \dots, X_p)$, $p \in \mathbb{N}$, may affect a response variable $Y$. This includes perfect dependence of $Y$ on $\mathbf{X}$ and independence between $\mathbf{X}$ 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 $\mathbf{X}$. This dimension reduction principle allows these measures to be estimated via the powerful nearest neighbor based estimation principle introduced in [4]. 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 $(\mathbf{X},Y)$, into the Markov product $(Y,Y')$.