Recently, Bjøru et al. proposed a novel divide-and-conquer algorithm for bounding counterfactual probabilities in structural causal models (SCMs). They assumed that the SCMs were learned from purely observational data, leading to an imprecise characterization of the marginal distributions of exogenous variables. Their method leveraged the canonical representation of structural equations to decompose a general SCM with high-cardinality exogenous variables into a set of sub-models with low-cardinality exogenous variables. These sub-models had precise marginals over the exogenous variables and therefore admitted efficient exact inference. The aggregated results were used to bound counterfactual probabilities in the original model. The approach was developed for Markovian models, where each exogenous variable affects only a single endogenous variable. In this paper, we investigate extending the methodology to \textit{semi-Markovian} SCMs, where exogenous variables may influence multiple endogenous variables. Such models are capable of representing confounding relationships that Markovian models cannot. We illustrate the challenges of this extension using a minimal example, which motivates a set of alternative solution strategies. These strategies are evaluated both theoretically and through a computational study.

The causal effect is defined by Pearl's do operation as a probability distribution over observed data in the way that it is altered from one which generates data originally [Pearl 1995; Pearl 2009]. When dealing with causal effects in realworld problems, it is also necessary to take into account unobserved variables that is not included in data. In general, causal effects are counterfactual probability distributions that differ from data generating systems in the real world. When we consider the existence of unobserved data, it becomes a problem if it can be determined by observed data available. That is, we need to consider the identifiability of causal effects in this case. This problem has recently been resolved to a certain extent [Tian and Pearl 2002; Shpitser and Pearl 2006; Shpitser and Pearl 2012].

The validity OF a causal model can be tested ONLY IF the model imposes constraints ON the probability distribution that governs the generated data. IN the presence OF unmeasured variables, causal models may impose two types OF constraints : conditional independencies, AS READ through the d - separation criterion, AND functional constraints, FOR which no general criterion IS available.This paper offers a systematic way OF identifying functional constraints AND, thus, facilitates the task OF testing causal models AS well AS inferring such models FROM data.

We present a class of inequality constraints on the set of distributions induced by local interventions on variables governed by a causal Bayesian network, in which some of the variables remain unmeasured. We derive bounds on causal effects that are not directly measured in randomized experiments. We derive instrumental inequality type of constraints on nonexperimental distributions. The results have applications in testing causal models with observational or experimental data.