Causal Representation & Reasoning

• Causal representation and reasoning are methods that model variables and direct influences using Bayesian networks and DAGs to uncover causal structures.
• They support diverse inference tasks such as prediction, explanation, intervention, and counterfactual analysis by leveraging conditional independence properties.
• This framework integrates developmental insights and algorithmic advances to provide both theoretical and practical foundations for causal learning.

Causal representation and reasoning constitute a foundational paradigm in both artificial intelligence and cognitive science for describing, inferring, and learning about the mechanisms that underlie observable phenomena. At its core, causal representation involves modeling variables and their direct causal dependencies (typically as a directed acyclic graph), while causal reasoning denotes the various forms of inference—including prediction, explanation, intervention, and counterfactual analysis—that such a model supports. This area integrates insights from probabilistic graphical models, Bayesian inference, formal logic, psychology, and machine learning, allowing for rigorous statistical accounts of how agents (biological or artificial) can discover, update, and utilize causal knowledge.

1. Probabilistic Causal Graphs and Independence Structure

The probabilistic causal graph model, often formalized as a Bayesian network, is the canonical approach to representing causality in a system of variables $U$. Each node in a directed acyclic graph (DAG) represents a variable, and each directed edge denotes a direct causal influence. The associated joint probability distribution $P$ is assumed to satisfy the Markov condition—for each variable $X$, given its parents, $X$ is conditionally independent of all non-descendant variables:

$$X \perp \text{NonDescendants}(X) \mid \text{Parents}(X).$$

This property, colloquially termed "shielding," expresses that once the direct causes of an effect are known, additional information about other variables is irrelevant to predicting that effect.

Additionally, the faithfulness condition holds if all conditional independencies in $P$ are exactly those entailed by the DAG structure—no more, no less. Together, these properties ensure that conditional independence patterns in data can, in principle, be interpreted as reflections of the underlying causal architecture.

To formalize direct causal influence, Pearl and Verma’s condition states that $X$ causally affects $Y$ if there exists $Z$ and a context $S_{XY}$ such that: $$\neg I(Z, Y \mid S_{XY}), \quad I(Z, Y \mid S_{XY} \cup \{X\})$$ meaning $Z$ and $Y$ are dependent in the context $S_{XY}$, but become independent upon conditioning on $X$.

2. Bayesian Networks and Human Causal Reasoning

The operations of Bayesian networks are not merely technical constructs but provide a close descriptive account of human causal reasoning. Empirical studies affirm that people dynamically activate fragments of a larger causal structure and perform belief updating—remarkably consistent with the sequential application of Bayes’ rule along causal links in a Bayesian network.

A notable psychological phenomenon supporting this correspondence is discounting: the presence of one cause for an effect diminishes the subjective probability assigned to alternative causes. Laboratory studies indicate that humans, in both frequency and magnitude, perform causal discounting in a manner quantitatively consistent with Bayesian inference algorithms propagating evidence in a DAG. This effect is directly attributable to the Markov condition in Bayesian networks, wherein knowledge of one direct cause "shields off" other possible causes from influencing the inferred probability of the effect.

3. Learning Causal Structure: Developmental and Algorithmic Parallels

Empirical developmental studies, especially those by Piaget and Inhelder, reveal that children’s acquisition of causal knowledge progresses through increasingly sophisticated sensitivity to dependency and conditional independence structures among events. For example, young infants may initially learn two-variable associations ("pull string → toy shakes") but only later, after accumulating experience, discover more complex relationships (identifying intermediary variables mediating an effect).

This developmental trajectory mirrors the algorithmic form of causal discovery in Bayesian networks, where conditional independence tests (of the sort $I(Z, Y | S)$ and its negation) serve as the principal signal for orienting and structuring causal graphs. In essence, children’s causal learning exploits the same statistical regularities—independencies and dependencies—that form the computational heart of structure learning algorithms (e.g., the PC algorithm and its variants).

The learning of structure thus arises from observing which variables co-vary directly and which become conditionally independent upon controlling for other variables, directly aligning with algorithmic advances in graphical causal discovery.

4. Subjectivity and Statistical Nature of Causality

A central philosophical position advanced is the subjective definition of causality: causality is not an objective property of the world as such, but a relationship inferred by an observer on the basis of observed statistical regularities. In the formalism adopted—essentially Pearl and Verma's definition—causality for an individual is entirely determined by the statistical record of dependencies and independencies among variables as encountered by the observer:

• $X$ is a cause of $Y$ if there exists some $Z$ and context $S_{XY}$ such that the dependency structure above holds in the observed data.

This perspective stands in contrast to scientific or philosophical notions of causality grounded in mechanistic or metaphysical invariance, instead emphasizing that causal representations are a function of the agent's perceptual and cognitive filters. From a practical/algorithmic standpoint, this subjectivity does not detract from predictive or inferential power—it simply foregrounds the dependence of any causal graph on the observational scope and statistical configuration available to the learner.

5. Empirical and Theoretical Methods for Testing Causal Reasoning Theories

Multiple empirical strategies are proposed to evaluate both the structure and processes by which humans reason about causality:

• Subjective Probability Testing: Participants numerically estimate probabilities for components of a causal scenario, and these are compared against the normative (Bayesian) predictions. Successes, as in discounting experiments, suggest that people reason in line with probabilistic causal graphs.
• Prime-Probe Reaction Time Measures: By measuring how quickly individuals respond to tasks involving pairs of causal concepts, researchers can test whether the mental traversal of causal links mimics the network propagation schemes in Bayesian inference.
• Controlled Artificial Environments: Immersive or computer-simulated settings enable observation of the precise conditions triggering causal learning or inference, and can illuminate the timing and nature of belief updating in response to new dependencies or independencies.

These empirical paradigms are coupled with a theoretical imperative: to develop algorithms that match the sophistication and flexibility of human causal learning, potentially extending Bayesian network learning algorithms to operate without pre-fixed variable sets and to more flexibly adapt to observed regularities.

6. Integration and Implications

The probabilistic causal graph model systematically accounts for both the representation of causality—via the encoding of influences and independencies in a DAG with a joint distribution—and the processes of reasoning and structure learning—by aligning conditional independence algorithms with observed human behavior. The framework’s explanatory adequacy is supported by:

• Developmental evidence (Piaget's stages and observation of learning from conditional dependencies).
• Psychological discounting phenomena matching normative Bayesian reasoning.
• Practical methods for experimental assessment and theoretical expansion.

This synthesis not only guides further research in psychology and AI but also provides clear criteria for distinguishing causal from non-causal relationships in real-world data, highlighting the power of conditional independence and subjective statistical inference as organizing principles in the science of causality.