Graphical models use graphs to represent conditional independence structure
in the distribution of a random vector. In stochastic processes, graphs may
represent so-called local independence or conditional Granger causality. Under
some regularity conditions, a local independence graph implies a set of
independences using a graphical criterion known as δ-separation, or
using its generalization, μ-separation. This is a stochastic process
analogue of d-separation in DAGs. However, there may be more independences
than implied by this graph and this is a violation of so-called faithfulness.
We characterize faithfulness in local independence graphs and give a method to
construct a faithful graph from any local independence model such that the
output equals the true graph when Markov and faithfulness assumptions hold. We
discuss various assumptions that are weaker than faithfulness, and we explore
different structure learning algorithms and their properties under varying
assumptions.Comment: 17 pages, 3 figure