Self-consistency: a fundamental concept in statistics

Abstract

The term "self-consistency" was introduced in 1989 by Hastie and Stuetzle to describe the property that each point on a smooth curve or surface is the mean of all points that project orthogonally onto it. We generalize this concept to self-consistent random vectors: a random vector Y is self-consistent for X if $\mathscr{E}[X|Y] = Y$ almost surely. This allows us to construct a unified theoretical basis for principal components, principal curves and surfaces, principal points, principal variables, principal modes of variation and other statistical methods. We provide some general results on self-consistent random variables, give examples, show relationships between the various methods, discuss a related notion of self-consistent estimators and suggest directions for future research.