Causal Discovery

These algorithms use statistical tests for conditional independence to constrain the space of possible causal graphs. The most prominent is the PC algorithm (named after its creators, Peter Spirtes and Clark Glymour).

• Core Mechanism: Systematically tests for conditional independencies (e.g., X ⊥ Y | Z) in the data.
• Graphical Rule: Applies the d-separation criterion to map statistical independencies to graphical structures.
• Output: Produces a Partially Directed Acyclic Graph (PDAG), representing a Markov equivalence class of DAGs that share the same conditional independence relationships.
• Key Assumption: Relies heavily on the Causal Faithfulness and Causal Markov conditions.

These algorithms treat causal discovery as an optimization problem, searching for the graph structure that best fits the data according to a predefined scoring function.

• Core Mechanism: Defines a score function (e.g., Bayesian Information Criterion, Minimum Description Length) that balances model fit with complexity.
• Search Strategy: Employs heuristic search (e.g., greedy hill-climbing, tabu search) over the space of DAGs to find the highest-scoring graph.
• Output: Typically aims to find a single Directed Acyclic Graph (DAG) that maximizes the score.
• Example: The Greedy Equivalence Search (GES) algorithm starts with an empty graph and iteratively adds or deletes edges to improve the score, operating over equivalence classes for efficiency.

This approach assumes specific functional relationships between causes and effects, most commonly using Additive Noise Models (ANMs). It leverages asymmetry in the data generation process.

• Core Mechanism: Assumes each variable is a function of its parents plus independent noise: X_i = f(PA_i) + N_i.
• Identifiability: Under certain conditions (e.g., nonlinear f or non-Gaussian noise), the true causal direction becomes identifiable from observational data alone.
• Method: Tests for independence between the hypothesized cause and the residual noise of the hypothesized effect. The direction where the residual is independent is preferred.
• Algorithms: LiNGAM (Linear Non-Gaussian Acyclic Model) is a seminal algorithm in this family, assuming linear functions with non-Gaussian noise.

Focused on temporal data, these methods leverage the arrow of time: a cause must precede its effect. Granger causality is a foundational, though not strictly causal, concept in this domain.

• Granger Causality Definition: A variable X 'Granger-causes' Y if past values of X contain statistically significant information for predicting Y that is not contained in past values of Y alone.
• Limitation: Granger causality detects predictive precedence, which can be confounded by latent common causes. It is more accurately termed 'Granger prediction'.
• Modern Extensions: Algorithms like PCMCI (Peter-Clark Momentary Conditional Independence) extend constraint-based methods to time series, robustly handling autocorrelation and high-dimensional sets of variables.

Recent approaches leverage deep learning to perform causal discovery by making the graph structure a differentiable component of a neural network model.

• Core Mechanism: Represents the causal graph as an adjacency matrix where entries are continuous parameters. A neural network learns to predict data based on this graph.
• Optimization: Uses gradient descent to simultaneously optimize the continuous graph parameters and the neural network weights, often with a sparsity penalty (L1 regularization) on the graph.
• Algorithm: NOTEARS (Non-combinatorial Optimization via Trace Exponential and Augmented lagRangian for Structure learning) is a landmark method that formulates the acyclicity constraint as a smooth, differentiable function.
• Benefit: Scales better to high-dimensional data than traditional combinatorial search methods.

All causal discovery algorithms rest on foundational assumptions, and their outputs are only as valid as these assumptions hold.

• Causal Sufficiency: Assumes no unmeasured common causes (latent confounders) of the observed variables. Violation leads to missing edges or incorrect directions.
• Faithfulness & Markov Conditions: Link the causal graph's structure to the statistical independencies in the data. Faithfulness is violated if independencies arise from precise parameter cancelations.
• Acyclicity: Most methods assume no feedback loops (a Directed Acyclic Graph). Methods for cyclic graphs (e.g., from time series or equilibrium data) are more complex.
• Data Distribution: Functional methods (like LiNGAM) require specific distributional assumptions (non-Gaussianity, nonlinearity) for identifiability.
• Output Ambiguity: Often returns an equivalence class of graphs, not a unique DAG, highlighting the inherent limitations of learning causation from observation alone.

A Structural Causal Model (SCM) is the formal mathematical framework underpinning causal reasoning. It represents causal relationships as a system of structural equations, typically visualized as a causal graph (a Directed Acyclic Graph). Each variable is defined as a function of its direct causes and an independent noise term. SCMs enable precise reasoning about interventions (the do-operator) and counterfactuals, providing the semantics for moving beyond statistical association.

Causal inference is the process of estimating the quantitative effect of a specific intervention or treatment from data. While causal discovery aims to learn the graph structure, causal inference uses a known or assumed structure to answer "what if" questions. Core methods include:

• Potential Outcomes Framework: Compares outcomes under treatment vs. control.
• Do-Calculus: A set of rules for translating interventional queries into observational probabilities.
• Estimation Techniques: Such as propensity score matching and inverse probability weighting to adjust for confounding.

A causal graph is a visual and mathematical representation of a system's causal assumptions. It is a Directed Acyclic Graph (DAG) where:

• Nodes represent variables.
• Directed Edges represent direct causal relationships (X → Y means X is a direct cause of Y). These graphs encode conditional independence relationships via d-separation. They are essential for identifying which statistical adjustments (conditioning on which variables) are necessary and sufficient for unbiased causal estimation, using criteria like the backdoor criterion and frontdoor criterion.

Do-calculus is a complete set of three inference rules developed by Judea Pearl for causal reasoning. It allows researchers to compute the probabilities of interventions (e.g., P(Y | do(X))) from purely observational data, provided the causal graph is known. The rules formally manipulate expressions containing the do-operator, enabling the identification of causal effects even in the presence of unobserved confounders in certain graph structures. It is the mathematical engine that powers many causal inference algorithms.

Causal identifiability is the fundamental property that determines whether a causal quantity of interest (like the Average Treatment Effect) can be uniquely computed from the available data under a given causal model. It answers the question: "Can we estimate this effect without bias, given our assumptions?" Identifiability often hinges on satisfying specific graphical conditions (e.g., no unmeasured confounding for the backdoor path). If an effect is not identifiable, no statistical method can reliably estimate it from the observed data alone.

The causal hierarchy, or ladder of causation, is a three-level framework that categorizes the types of questions an AI system can answer:

1. Association (Seeing): "What is?" Observing correlations. (Machine Learning / Statistics)
2. Intervention (Doing): "What if?" Predicting effects of actions. (Causal Inference)
3. Counterfactual (Imagining): "Why?" Reasoning about what would have happened under different circumstances. Each level requires more sophisticated causal knowledge and is strictly above the one before it. Causal discovery aims to provide the models needed to climb this ladder.