Selectivity in Probabilistic Causality: Drawing Arrows from Inputs to Stochastic Outputs

TL;DR

This paper introduces a criterion and linear programming-based tests to determine which inputs influence specific outputs in stochastic systems, with broad applications in modeling and psychological research.

Contribution

It presents the Joint Distribution Criterion as a necessary and sufficient condition for selectivity, linking it to linear feasibility tests and generating a wide range of necessary influence tests.

Findings

Provides a linear programming test for influence patterns

Introduces distance-type tests based on triangle inequalities

Offers a unifying criterion for selective influences

Abstract

Given a set of several inputs into a system (e.g., independent variables characterizing stimuli) and a set of several stochastically non-independent outputs (e.g., random variables describing different aspects of responses), how can one determine, for each of the outputs, which of the inputs it is influenced by? The problem has applications ranging from modeling pairwise comparisons to reconstructing mental processing architectures to conjoint testing. A necessary and sufficient condition for a given pattern of selective influences is provided by the Joint Distribution Criterion, according to which the problem of "what influences what" is equivalent to that of the existence of a joint distribution for a certain set of random variables. For inputs and outputs with finite sets of values this criterion translates into a test of consistency of a certain system of linear equations and inequalities (Linear Feasibility Test) which can be performed by means of linear programming. The Joint Distribution Criterion also leads to a metatheoretical principle for generating a broad class of necessary conditions (tests) for diagrams of selective influences. Among them is the class of distance-type tests based on the observation that certain functionals on jointly distributed random variables satisfy triangle inequality.