Causal Bayesian Networks

• Causal Bayesian networks are probabilistic graphical models defined by a directed acyclic graph where nodes represent variables and edges denote direct causal influences.
• They facilitate evaluation of interventions using do-calculus, replacing node mechanisms to perform counterfactual analysis and answer interventional queries.
• They are commonly discovered from data via constraint-based or score-based methods, though challenges like Markov equivalence and latent confounders can limit identifiability.

A causal Bayesian network (CBN) is a probabilistic graphical model defined by a directed acyclic graph (DAG) whose nodes correspond to random variables and whose edges represent direct causal relationships. Each node is associated with a conditional probability distribution (typically a conditional probability table, or CPT) given its parents in the graph. The joint distribution over all variables factorizes according to the DAG structure, and the directed edges admit interventions that correspond to manipulating causal mechanisms, enabling evaluation of counterfactuals and interventional queries.

1. Foundations: Mechanism-Based Semantics and DAG Factorization

The mechanism-based view of causality, originally formalized in structural equation modeling (SEM) by Simon, serves as the rigorous foundation for causal Bayesian networks. Here, each endogenous variable $X_i$ is defined by a structural equation $X_i = f_i(Pa_i, \epsilon_i)$, where $Pa_i$ denotes the direct causes (parents) of $X_i$ in the system and $\epsilon_i$ is an exogenous, independent noise term. Exogenous variables are not caused by others in the model and are thus fixed outside the system, whereas endogenous variables result from the system's mechanisms.

A Bayesian belief network, or Bayesian network, is specified by:

• A finite set of discrete variables $X_1, \ldots, X_n$.
• A DAG $G$ whose nodes are these variables.
• For each node $X_i$, a CPT $P(X_i|Pa_i)$.

Under the Markov property, the joint distribution is

$$P(X_1, \ldots, X_n) = \prod_{i=1}^n P(X_i \mid Pa_i).$$

The parents $Pa_i$ correspond to the minimal set of variables upon which $X_i$ probabilistically depends, and, under mechanism-based semantics, they coincide with the direct causes in SEM (Druzdzel et al., 2013).

2. When Is a Bayesian Network Causal?

Acyclic Bayesian networks admit a causal interpretation when the following semantic criteria are satisfied (Druzdzel et al., 2013):

• (C1) Mechanism-based condition: For every node $X_i$, the collection $\{X_i\} \cup Pa_i$ matches precisely the variables involved in a real-world mechanism, i.e., there exists a structural equation $X_i = f_i(Pa_i, \epsilon_i)$ whose noise term $\epsilon_i$ is mutually independent from other exogenous terms, with $P(X_i|Pa_i)$ arising from this mechanism.
• (C2) Exogeneity condition: Nodes with $Pa_i = \emptyset$ must truly be exogenous, i.e., set by nature or externally fixed.

When these conditions hold and the DAG is acyclic, the network's directed edges correspond to genuine causal dependencies. Interventions (using Pearl's do-operator) are modeled by replacing a structural equation for a variable with a fixed assignment and propagating the effect only downstream along the edges (Druzdzel et al., 2013). The correspondence between the DAG and the causal ordering of the underlying SEM is guaranteed when the noise terms are independent and the structural equations possess causal directionality.

3. Intervention, Do-Calculus, and Causal Asymmetry

Interventions in a CBN sever the normal mechanism of a node and replace it with a fixed value (the do-operator formalism). Mathematically, for $do(X_j = x^*)$, the CPT for $X_j$ is replaced by a degenerate distribution enforcing $X_j = x^*$, and only edges outgoing from $X_j$ propagate changes. The resulting distribution is computed by truncating the original factorization so that $X_j$ is fixed, with all other factors unchanged.

A crucial distinction between causality and probabilistic inference is this causal asymmetry: in a purely probabilistic network, Bayes' theorem allows edges to be reversed and conditional dependencies to be refactored arbitrarily, but in a mechanism-based network, the directions are generally not reversible without changing the system's mechanisms or noise structures (Druzdzel et al., 2013).

4. Conditional Independence, D-Separation, and Faithfulness

In CBNs, the edges encode not only joint factorization but also a precise set of conditional independence (CI) relations determined by d-separation in the DAG. The Markov condition states: each node is independent of its nondescendants given its parents. Faithfulness, or stability, requires that all and only the CIs present in the distribution are those implied by d-separation (Morris et al., 2013).

For three disjoint sets $A,B,C$, $A$ is d-separated from $B$ given $C$ if every path between a node in $A$ and a node in $B$ is blocked by $C$ using the usual collider and chain/fork blocking rules. If a distribution is Markov and faithful to the graph, the encoded CI structure is complete.

5. Structure Learning and Causal Discovery from Data

Learning a CBN's structure from data hinges on detecting conditional independencies (constraint-based methods, e.g., PC algorithm) or optimizing a goodness-of-fit score while penalizing model complexity (score-based methods with BDe or BIC scores). For score-based Bayesian learning, five key assumptions are standard (Heckerman, 2013):

1. Parameter independence (priors over parameters for each CPT are jointly independent).
2. Parameter modularity (priors for families with identical parent sets are identical across structures).
3. Likelihood equivalence (Markov-equivalent DAGs yield identical marginal likelihood for observational data).
4. Mechanism independence (the causal mechanisms for different variables are independent).
5. Component independence (within a single CPT, the parameters for different parent configurations are a priori independent).

Under these conditions, the marginal likelihood of data given a DAG structure has a closed-form (multivariate Dirichlet-multinomial/BDe score), and the posterior over structures can be computed efficiently (Heckerman, 2013). When interventional or mixed sampled data are available, intervention nodes or selection indicators are introduced as "explicit" variables in the DAG, and the Bayesian machinery generalizes naturally (Cooper, 2013).

For non-random (selection-biased) sampling, an explicit selection variable $S$ is included, and the data likelihood and model structure must incorporate the role of $S$, requiring summing/marginalizing over unobserved variables and ancestors of $S$; efficient computation may be possible in particular DAG structures (Cooper, 2013).

6. Limitations, Identifiability, and Structural Distinctions

CBNs, even when appropriately specified, are limited by several identifiability issues:

• Markov equivalence: Without interventional data or additional assumptions, only the Markov equivalence class (set of DAGs encoding the same CI structure) is identifiable from strictly observational data (Morris et al., 2013, Heckerman, 2013).
• Collider and directionality resolution: Certain structural motifs (e.g., colliders) are statistically harder to detect and require larger sample sizes or additional constraints to orient edges reliably (White et al., 2018). Algorithms' performance in correctly orienting edges in practice depends strongly on network size, density, sample size, and available prior knowledge (Butcher et al., 2020).
• Latent confounders: Presence of unobserved variables can introduce dependencies that are not explainable within the observed variable DAG, necessitating extended graphical representations (e.g., mixed graphs with bidirected edges) or mixture models with latent classes to restore identifiability (Gordon et al., 2021, Gonzales et al., 20 Aug 2024).

7. Applications and Theoretical Significance

Causal Bayesian networks provide the formal foundation for do-calculus, intervention-based reasoning, and structure learning in a variety of domains—biological systems, socioecological modeling, safety analysis in engineering, and beyond (White et al., 2018, Gansch et al., 26 May 2025, Cabañas et al., 18 Jan 2024). By explicitly characterizing when the DAG structure corresponds to underlying mechanisms, CBNs unify the graphical, algebraic, and algorithmic paradigms of causal inference, serving as the workhorse for both theoretical analysis and applied causal discovery (Druzdzel et al., 2013).

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References:

• "Causality in Bayesian Belief Networks" (Druzdzel et al., 2013)
• "The Cognitive Processing of Causal Knowledge" (Morris et al., 2013)
• "A Bayesian Method for Causal Modeling and Discovery Under Selection" (Cooper, 2013)
• "A Bayesian Approach to Learning Causal Networks" (Heckerman, 2013)
• "Causal Queries from Observational Data in Biological Systems via Bayesian Networks: An Empirical Study in Small Networks" (White et al., 2018)