Causal graphs, composable stochastic processes and conditional independence

Wojciech Niemiro¹

¹ Institute of Applied Mathematics and Mechanics University of Warsaw Warszawa, Poland

Applicationes Mathematicae, Tome 51 (2024) no. 2, pp. 109-130.

Résumé

We consider multivariate stochastic processes with causal relations between their components modelled by directed graphs with possible cycles. Our aim is to express conditional independence relations for such processes in terms of separability properties of the underlying graphs. This line of study is quite classical and was initiated in the seminal paper of Pearl (1985), then extended to point processes by Didelez (2007, 2008) and to time series by Eichler (2007) and Eichler and Didelez (2007). In our paper we provide a unifying view and fill in certain gaps. We define a class of models called composable random elements (CRE) which encompasses usual Bayesian networks (BN), dynamic BNs (DBN), continuous time BNs (CTBN) and marked point processes. We show that key results known in the classical setup of directed acyclic graphs (DAG) can be generalised to CREs and remain valid also for graphs containing cycles. For CTBNs, we prove a new theorem that characterises independence between the future of one subprocess and the past of another given the past of a third subprocess. Our paper also tackles causal (interventional) conditional independence relations, strengthening and generalising results of Ay and Polani (2008).

Détail

DOI : 10.4064/am2511-10-2024

Keywords: consider multivariate stochastic processes causal relations between their components modelled directed graphs possible cycles express conditional independence relations processes terms separability properties underlying graphs line study quite classical initiated seminal paper pearl extended point processes didelez time series eichler eichler didelez paper provide unifying view fill certain gaps define class models called composable random elements cre which encompasses usual bayesian networks dynamic bns dbn continuous time bns ctbn marked point processes key results known classical setup directed acyclic graphs dag generalised cres remain valid graphs containing cycles ctbns prove theorem characterises independence between future subprocess past another given past third subprocess paper tackles causal interventional conditional independence relations strengthening generalising results polani