Causal Bayesian Networks

• Causal Bayesian Networks are graphical models that represent causal relationships using directed acyclic graphs and factorized conditional probabilities.
• They employ do-calculus to evaluate interventions, counterfactuals, and derive causal effects from both experimental and observational data.
• CBNs support structure learning via constraint-based, score-based, and hybrid methods, with applications spanning genomics, epidemiology, and manufacturing.

A Causal Bayesian Network (CBN) is a graphical probabilistic model that represents causal relationships among random variables using a directed acyclic graph (DAG) and a set of conditional probability distributions that jointly specify the data-generating process. Each edge in the graph conveys direct causal influence, enabling counterfactual and interventional reasoning via the machinery of the do-operator and do-calculus. CBNs are foundational in diverse scientific, engineering, and medical fields for both causal discovery and as tools for evaluating the effects of hypothetical interventions.

1. Formal Definition and Structural Properties

A CBN is formally defined as a pair $(G, P)$, where $G=(V,E)$ is a directed acyclic graph whose nodes $X_1, ..., X_n$ are random variables, and $P$ is a joint distribution over $(X_1, ..., X_n)$ that factorizes as

$$P(X_1, ..., X_n) = \prod_{i=1}^n P(X_i \mid \mathrm{Pa}_G(X_i))$$

with $\mathrm{Pa}_G(X_i)$ being the set of parents of $X_i$ in $G$ (Morris et al., 2013).

The core graphical criterion is d-separation, which determines whether sets of variables are conditionally independent, as expressed in the Markov property:

• Each $X_i$ is conditionally independent of its non-descendants given its parents.
• d-separation characterizes all and only those conditional independencies implied by $G$; these must be reflected in $P$ for the CBN to be Markov to $G$.

The faithfulness condition strengthens this correspondence: a distribution $P$ is faithful to $G$ if all conditional independencies present in $P$ are reflected via d-separation in $G$ (Lin et al., 2018). Markov equivalence classes—sets of DAGs inducing the same set of d-separation relations—arise naturally under observational data, constraining what can be learned from data alone.

2. Causal Semantics and Interventional Calculus

CBNs differ from generic Bayesian networks through the explicit causal interpretation of directed edges and their capacity to formally model interventions. The do-operator, $do(X=x)$, severs all incoming edges into $X$ and replaces its conditional probability with a degenerate distribution assigning probability 1 to $X=x$ (Galhotra et al., 23 May 2024). The resulting interventional distribution is given by the truncated factorization: $$P(Y \mid do(X = x)) = \sum_{z} P(Y \mid X = x, z) P(z)$$ for appropriate $z$, under mild independence and autonomy assumptions.

General intervention calculus is guided by Pearl's do-calculus, which provides sound rules for manipulating and identifying interventional distributions under various graphical conditions (White et al., 2018, Galhotra et al., 23 May 2024). Back-door and front-door adjustment formulas enable identifiability of causal effects from purely observational data, provided the relevant graphical criteria are satisfied.

3. Structure Learning and Model Selection

Learning CBN structure from data operates under several paradigms, each with theoretical and practical trade-offs:

• Constraint-based methods (e.g., PC algorithm, IAMB): Identify the essential graph skeleton and edge orientations by testing conditional independencies, exploiting d-separation logic. Algorithms such as PC build the undirected graph, prune edges based on CI tests, and orient edges within equivalence classes using orientation rules (Morris et al., 2013, White et al., 2018, Kitson et al., 2019).
• Score-based methods: Search for the DAG that maximizes a decomposable score, typically BIC or BDeu, over the space of acyclic directed graphs, often employing greedy hill-climbing and tabu search (Kitson et al., 2019, Wehner et al., 20 Jan 2024).
• Hybrid and Constrained methods: Restrict the search space using constraint-based candidate parents, then apply score-based optimization to the reduced set. Domain knowledge can be incorporated via blacklists/whitelists or prior graph structure, yielding more plausible and robust models (Kitson et al., 2019, Wehner et al., 20 Jan 2024, Beaumont et al., 2017, Kitson et al., 2023).

Structure learning under latent confounding requires specialized algorithms that output Partial Ancestral Graphs or that post-process learned structures to detect patterns consistent with unobserved confounders (Gonzales et al., 20 Aug 2024, Chobtham et al., 2020).

For sparse networks, recent results demonstrate that exact structure discovery can be performed in polynomial time when the optimal DAG is a matching or when components are logarithmically bounded (Rios et al., 21 Jun 2024).

4. Inference, Interventions, and Causal Query Answering

Once a CBN's structure and parameters are estimated, a broad range of inferences are supported:

• Interventional distributions: Computing $P(Y \mid do(X=x))$ by modifying the graph and conditional probabilities as dictated by the intervention (Galhotra et al., 23 May 2024, Gansch et al., 26 May 2025).
• Counterfactuals: Evaluated via abduction–action–prediction, requiring both abduction to infer latent variables (or exogenous noise), then applying the intervention and marginalizing to predict $Y$ (Zahoor et al., 27 Jan 2025).
• Causal effect and pathway analysis: Quantifying average or relative causal effects, path-specific effects, or decomposing effects by alternative intervention regimes (Gansch et al., 26 May 2025).

Testable model implications can be derived: in fully observed CBNs, the joint distribution under interventions is subject to explicit polynomial equality constraints, computable via algebraic-geometric methods (implicitization), particularly in the presence of hidden variables (Kang et al., 2012).

5. Specialized CBN Methodologies and Applications

The CBN formalism has been extended and tailored for various specialized contexts:

• Expert/interactive constraint integration: Knowledge graphs and active human-in-the-loop querying are used for structure learning in high-stakes settings (e.g., manufacturing root cause analysis), offering improved search efficiency and spurious edge reduction (Wehner et al., 20 Jan 2024, Kitson et al., 2023).
• Controllability analysis: The minimal set of driver nodes for probabilistic structural controllability can be precisely identified via systematic backward-chaining on the graph (Nobandegani et al., 2015).
• Interventional identifiability metrics: Key measures such as the probability of necessity, sufficiency, and necessity-and-sufficiency are identified from observational data when conditions of autonomy and appropriate independence hold (Galhotra et al., 23 May 2024).
• Constrained Bayesian Networks: CBNs with symbolic or interval-valued CPTs and real-valued constraints permit robust inference even with underdetermined or model-uncertain domains, solved via SMT and global optimization (Beaumont et al., 2017).
• Explanation and interpretability: Causal explanation trees select variables according to their information flow (in the interventional sense) to the explanandum, thus providing explanations aligned with the do-calculus semantics (Nielsen et al., 2012).

Applications span genomics (White et al., 2018), epidemiology (Zahoor et al., 27 Jan 2025), manufacturing (Wehner et al., 20 Jan 2024), safety analysis of AI/ML-based perception systems (Gansch et al., 26 May 2025), and large-scale demographic health surveys (Kitson et al., 2019).

6. Foundations, Limitations, and Active Controversies

The causal content of a CBN hinges on a mapping between real-world actions and model interventions. Recent work demonstrates that without a precise interpretation of this mapping, the distinction between causal and observational fits is obliterated: every CBN fitting the data is trivially valid for all interventions, making the model unfalsifiable (Jørgensen et al., 31 Jan 2025). Any non-circular interpretation that permits falsification must drop at least one desirable property (e.g., invariance to Markov equivalence, etc.). This reframes much of causal discovery, representation learning, and abstraction as inextricably bound to explicit specifications of the action–intervention correspondence.

Additionally, identifiability and learnability are fundamentally limited by latent confounding, faithfulness violations, and the computational hardness of discovering optimal structures in dense regimes (Rios et al., 21 Jun 2024, Lin et al., 2018, Gonzales et al., 20 Aug 2024).

A rigorous boundary is drawn: without faithfulness, no universally consistent algorithm exists that will recover structure everywhere; yet with faithfulness (or its necessity under optimal convergence desiderata), structure learning is maximally successful on topologically generic, non-pathological cases (Lin et al., 2018).

7. Algebraic, Statistical, and Computational Aspects

Analysis in CBNs leverages algebraic-geometric characterization, especially via polynomial equality constraints which fully capture the empirical content of the model, even when hidden variables are present. This allows both model testing (does the data satisfy the implied polynomial equalities?) and discrimination between candidate graphs via comparison of their ideals (Kang et al., 2012).

Score-based learning is computationally intractable in the general case (search space size $O(2^{n(n-1)/2})$ for $n$ variables). However, polynomial time structure identification is provably feasible in classes of sparse networks, and dynamic programming, path-search, and divide-and-conquer criteria can be applied for aggressive pruning (Rios et al., 21 Jun 2024).

Tables summarizing core CBN inference concepts:

Concept	Formula (LaTeX)	Context
Joint factorization	$$P(X_1, ..., X_n) = \prod_{i=1}^n P(X_i \mid \mathrm{Pa}_G(X_i))$$	Markov property
Do-intervention	$P(y \mid do(x)) = \sum_z P(y \mid x, z) P(z)$	Interventional query
Back-door adjustment	$P(y \mid do(x)) = \sum_z P(y \mid x,z)P(z)$ (if $Z$ satisfies crit.)	Causal identifiability
Faithfulness	$I(G) = I(P)$	Structure learnability
Polynomial constraint	$f(x_1, ..., x_N) = 0$ for $x_i = P(v|do(t))$	Model testing

In summary, Causal Bayesian Networks provide a rigorous mathematical framework for representing, learning, and reasoning about causality in probabilistic systems. Their power derives from the combination of graphical structure, probabilistic semantics, and interventional calculus, yet their application requires careful attention to the epistemological and computational assumptions that underlie causal inference from data.