Causal compositional models in valuation-based systems with examples in specific theories

• We show that Pearl's causal networks can be described using causal compositional models in the valuation-based systems.
• We introduce a novel notion of dominance that is necessary to describe the composition of valuations.
• We present an algebraic example of hidden variables elimination in the VBS framework.
• We provide numerical examples in probability, possibility, and Spohn's theories.

We show that Pearl's causal networks can be described using causal compositional models (CCMs) in the valuation-based systems (VBS) framework. One major advantage of using the VBS framework is that as VBS is a generalization of several uncertainty theories (e.g., probability theory, a version of possibility theory where combination is the product t-norm, Spohn's epistemic belief theory, and Dempster–Shafer belief function theory), CCMs, initially described in probability theory, are now described in all uncertainty calculi that fit in the VBS framework. We describe conditioning and interventions in CCMs. Another advantage of using CCMs in the VBS framework is that both conditioning and intervention can be easily described in an elegant and unifying algebraic way for the same CCM without having to do any graphical manipulations of the causal network. We describe how conditioning and intervention can be computed for a simple example with a hidden (unobservable) variable. Also, we illustrate the algebraic results using numerical examples in some of the specific uncertainty calculi mentioned above.