Conditional Independencies under the Algorithmic Independence of Conditionals

摘要

In this paper we analyze the relationship between faithfulnessand the more recent condition of algorithmic Independence ofConditionals (IC) with respect to the Conditional Independencies(CIs) they allow. Both conditions have been extensively used forcausal inference by refuting factorizations for which thecondition does not hold. Violation of faithfulness happens whenthere are CIs that do not follow from the Markov condition. Forthose CIs, non-trivial constraints among some parameters of theConditional Probability Distributions (CPDs) must hold. Whensuch a constraint is defined over parameters of different CPDs,we prove that IC is also violated unless the parameters have asimple description. To understand which non-Markovian CIs arepermitted we define a new condition closely related to IC: theIndependence from Product Constraints (IPC). The conditionreflects that CIs might be the result of specificparameterizations of individual CPDs but not from constraints onparameters of different CPDs. In that sense it is morerestrictive than IC: parameters may have a simple description.On the other hand, IC also excludes other forms of algorithmicdependencies between CPDs. Finally, we prove that on top of theCIs permitted by the Markov condition (faithfulness), IPC allowsnon-minimality, deterministic relations and what we calledproportional CPDs. These are the only cases in which a CIfollows from a specific parameterization of a single CPD. color="gray">