A general definition of influence between stochastic processes

Anne Gégout-Petit (1, 2) , Daniel Commenges (3, 4)

1 IMB - Institut de Mathématiques de Bordeaux
2 CQFD - Quality control and dynamic reliability
3 Epidémiologie et Biostatistique [Bordeaux]
4 ISPED - Institut de Santé Publique, d'Epidémiologie et de Développement

We extend the study of weak local conditional independence (WCLI) based on a measurability condition made by Commenges and Gégout-Petit (2009) to a larger class of processes that we call D'. We also give a definition related to the same concept based on certain likelihood processes, using the Girsanov theorem. Under certain conditions, the two definitions coincide on D'. These results may be used in causal models in that we define what may be the largest class of processes in which influences of one component of a stochastic process on another can be described without ambiguity. From WCLI we can contruct a concept of strong local conditional independence (SCLI). When WCLI does not hold, there is a direct influence while when SCLI does not hold there is direct or indirect influence. We investigate whether WCLI and SCLI can be defined via conventional independence conditions and find that this is the case for the latter but not for the former. Finally we recall that causal interpretation does not follow from mere mathematical definitions, but requires working with a good system and with the true probability.

Mots clés: Causality; causal influence; directed graphs; dynamical models; likelihood process; stochastic processes

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Anne Gégout-Petit, Daniel Commenges. A general definition of influence between stochastic processes. Lifetime Data Analysis, 2010, 16 (1), pp.33--44. ⟨10.1007/s10985-009-9131-7⟩. ⟨hal-00386649⟩