An Algebraic Framework for Bayes Nets of Time Series

Abstract

Graphical models in which every node holds a time-series were analysed in Caines et al. [13], [14] using the notion of lattice conditional independence (LCI) due to Anderson et al. [1], [2]. Under certain feedback free (or causality) conditions, LCI imposes a special zero structure on the stochastic realizations of those processes generated by state space systems; this structure comes directly from the transitive directed acyclic graph (TDAG) which is in one-to-one correspondence with the Boolean Hilbert lattice of the LCI formulation. In this paper, these properties of sets of stochastic processes are generalized to the setting of infinite Bayes nets of time series; this formulation contains as a special case conditionally independent processes embedded in chain recurrent spatial structures.