Graphical models of local independence in stochastic processes

Graphs are often used as representations of conditional independence structures of random vectors. In stochastic processes, one may use graphs to represent so-called local independence. 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 so-called 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 certain 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.

Deciding Markov equivalence of two directed mixed graphs is computationally hard, and we introduce a class of equivalence relations that are weaker than Markov equivalence, i.e., lead to larger equivalence classes. The weak equivalence classes enjoy many of the same properties as the Markov equivalence classes, and they provide a computationally feasible framework while retaining a clear interpretation. We discuss how this can be used for graphical modeling and causal structure learning based on local independence.