# Graphical Modeling of Local Independence in Dynamical Systems

Local independence is an asymmetric notion of independence which describes how a system of stochastic processes (e.g. point processes or diffusions) evolves over time. Let A, B, and C be three subsets of the coordinate processes of the stochastic system. Intuitively speaking, B is locally independent of A given C if at every point in time knowing the past of both A and C is not more informative about the present of B than knowing the past of C only. Directed graphs can be used to describe the local independence structure of the stochastic processes using a separation criterion which is analogous to d-separation. In such a local independence graph, each node represents an entire coordinate process rather than a single random variable.

In this talk, we will describe various properties of graphical models of local independence and then turn our attention to the case where the system is only partially observed, i.e. some coordinate processes are unobserved. In this case, one can use directed mixed graphs to describe the local independence structure of the observed coordinate processes. Several directed mixed graphs may describe the same local independence model, and therefore it is of interest to characterize such equivalence classes of directed mixed graphs. It turns out that directed mixed graphs satisfy a central maximality property which allows one to construct a simple graphical representation of an entire Markov equivalence class of marginalized local independence graphs. This is convenient as the equivalence class can be learned from data and its graphical representation concisely describes what underlying structure could have generated the observed local independencies.