  Causal background knowledge about the existence or the absence of causal
edges and paths is frequently encountered in observational studies. The shared
directed edges and links of a subclass of Markov equivalent DAGs refined due to
background knowledge can be represented by a causal maximally partially
directed acyclic graph (MPDAG). In this paper, we first provide a sound and
complete graphical characterization of causal MPDAGs and give a minimal
representation of a causal MPDAG. Then, we introduce a novel representation
called direct causal clause (DCC) to represent all types of causal background
knowledge in a unified form. Using DCCs, we study the consistency and
equivalency of causal background knowledge and show that any causal background
knowledge set can be equivalently decomposed into a causal MPDAG plus a minimal
residual set of DCCs. Polynomial-time algorithms are also provided for checking
the consistency, equivalency, and finding the decomposed MPDAG and residual
DCCs. Finally, with causal background knowledge, we prove a sufficient and
necessary condition to identify causal effects and surprisingly find that the
identifiability of causal effects only depends on the decomposed MPDAG. We also
develop a local IDA-type algorithm to estimate the possible values of an
unidentifiable effect. Simulations suggest that causal background knowledge can
significantly improve the identifiability of causal effects.
