Causal Discovery: Learning Causal Structure from Data

This notebook teaches causal discovery through interactive visualizations and incremental examples, building intuition about how algorithms infer causal structure from observational data.

Causal Discovery¶

Cell 1: The Core Problem: Correlation vs. Causation¶

Goal:

• Recognize that identical observational patterns can hide opposite causal implications

Plots:

• Two scatter plots side-by-side showing identical correlation patterns
• Both show X and Y with strong correlation (r = 0.8 to 0.99)
• Left plot labeled: “Chain: X → Y” with annotation “(if we intervene on X, Y changes)”
• Right plot labeled: “Reverse: Z → Y → X” with annotation “(intervening on X has no effect)”
• Summary annotations showing decision scenarios

Parameters:

• Correlation (r): Strength of correlation between variables (0.3 to 0.99)
• Mode: Toggle between observational vs. interventional reasoning

Key observations:

• Prediction and causation require fundamentally different reasoning
• Correlation alone cannot determine causal direction
• Same correlation pattern can hide opposite causal implications
  • This is the core problem: no amount of observational data alone distinguishes these structures
  • We need additional assumptions or interventions

Cell 2: Markov Equivalence: Three Indistinguishable Structures¶

Goal:

• Show that different DAGs encode identical conditional independencies
• Introduce the CPDAG as the best we can do with observational data alone
• Understand why causal discovery outputs equivalence classes, not unique DAGs

Plots:

• Three DAGs side-by-side: Chain (X → Y → Z), Reverse (Z → Y → X), Common Cause (X ← Y → Z)
• Below each DAG: list of implied conditional independencies
  • All three imply: X ⊥ Z | Y (X independent of Z given Y)
  • Distributions are mathematically identical
• Visual summary: CPDAG showing equivalence class with skeleton and ambiguous edge marks
• Histograms showing correlation patterns:
  • Marginal correlation X-Z: identical across all three structures
  • Conditional correlation X-Z|Y: near-zero for all

Parameters:

• Sample Size (N): Number of observations (50 to 5000)
  • Larger samples provide stronger statistical power for CI tests
  • But even infinite samples cannot distinguish Markov equivalent structures

Key observations:

• Observational data recover only conditional independence structure
• Edge directions in chains are fundamentally ambiguous
• CPDAG is the most informative output possible from observational data alone
• All three structures satisfy the same independencies and have identical covariance matrices

Cell 3: Why Direction Matters: Causal Effects via Intervention¶

Goal:

• Show concretely why edge direction determines causal effect
• Build intuition for why structure recovery is crucial for policy
• Demonstrate that same correlation leads to wildly different intervention outcomes

Plots:

• Three side-by-side panels showing counterfactual outcomes for each structure
• Scenario: intervene on X (set to high value) and observe effect on Z
  • Chain (X → Y → Z): large effect on Z (green bar)
  • Reverse (Z → Y → X): no effect on Z (red bar)
  • Common Cause (Y confounds both): no direct effect on Z (orange bar)
• Bar chart showing effect sizes with confidence bands
• Visualization of how effect propagates (or not) through the DAG

Parameters:

• Intervention Strength: Magnitude of change to X (0 to 3)
• Sample Size (N): Observations used for effect estimation (100 to 5000)

Key observations:

• Same observational correlation, wildly different causal effects
• Edge direction determines whether an intervention is effective
• Policy decisions depend critically on DAG structure
• Choosing the wrong DAG leads to ineffective or harmful interventions

Cell 4: The PC Algorithm: Learning From Conditional Independence Tests¶

Goal:

• Understand constraint-based discovery
• See step-by-step how PC uses conditional independence (CI) tests
• Build intuition for why CI tests are powerful but fallible

Plots:

• Animated visualization of PC algorithm on 4-5 variable DAG
• Left panel: evolving graph structure
  • Start with fully connected graph
  • Step-by-step edge removal as CI tests pass
  • Highlight separating sets
• Right panel: current test details
  • Test number and description
  • p-value and threshold comparison
  • Separating set information

Parameters:

• Alpha (CI threshold): Significance level for CI tests (0.001 to 0.2)
  • Lower alpha: stricter tests, keep more edges
  • Higher alpha: liberal tests, remove more edges
• Test Type: CI test method (Partial correlation, Gaussian G-squared, Conditional MI)
• Speed: Animation speed for step-by-step progression

Key observations:

• PC recovers skeleton via CI tests, then orients v-structures (colliders)
• Stronger distributional assumptions enable more powerful tests
• CI test errors propagate directly to DAG structure errors
• PC is sound in the large-sample limit, but CI tests are underpowered in finite samples

Cell 5: Score-Based Search: GES (Greedy Equivalence Search)¶

Goal:

• Introduce score-based discovery where algorithms optimize fit to data
• Show how BIC balances likelihood fit with model complexity
• Understand forward-backward search strategy

Plots:

• Left panel: animated DAG evolving during search
  • Starts empty (no edges)
  • Shows which edge is being added/removed at each step
  • Color-codes edges by score contribution (bright = high improvement)
  • Current BIC score displayed below graph
• Right panel: line plot of BIC score vs. iteration
  • Annotations for forward/backward phase transitions
  • Shows local maxima and convergence plateau

Parameters:

• Sample Size (N): Number of observations (50 to 10000)
  • BIC penalty proportional to log(N) for model complexity
  • Larger N allows more edges
• Regularization: Strength of sparsity penalty (0 to 2)
  • Higher regularization: fewer edges

Key observations:

• GES searches DAG space using greedy forward-backward heuristics
• BIC balances likelihood fit with model complexity
• Forward phase adds edges, backward removes low-value edges
• GES is guaranteed to recover the equivalence class in the limit
• Greedy search can get stuck in local optima; multiple random starts improve robustness

Cell 6: Non-Gaussian Methods: LiNGAM for Full Identifiability¶

Goal:

• Show how non-Gaussianity breaks directional symmetry
• Enable full DAG recovery instead of just equivalence classes
• Build intuition for when functional assumptions are powerful

Plots:

• Three scatter plots showing data clouds for X→Y relationships
  • Gaussian case: symmetric elliptical shapes (same covariance for both directions)
  • Non-Gaussian case: asymmetric shapes revealing causal direction
• Adjacency matrices showing recovered structure:
  • Matrix B where Xi = sum(bij * Xj) + ei
  • B is acyclic (uniquely oriented DAG)
• Histograms of noise distributions
  • Show non-Gaussianity metrics (heavy tails, skewness)
  • Highlight that ICA recovers independent non-Gaussian components

Parameters:

• Skewness: Non-Gaussianity of noise distribution (0 = Gaussian to 5 = heavy-tailed)
  • Gaussian: chains indistinguishable
  • Skewed: asymmetry reveals direction
• Signal-to-Noise Ratio: Ratio of signal to noise strength (0.1 to 0.99)

Key observations:

• Gaussian linear models are symmetric: X→Y and Y→X produce same distribution
• Non-Gaussianity breaks symmetry: LiNGAM exploits asymmetry to orient edges
• LiNGAM provides full DAG identification under linear non-Gaussian assumptions
• Jarque-Bera test can detect non-Gaussianity in data

Cell 7: Comparing Algorithms: Which One to Use?¶

Goal:

• Show outputs from PC, GES, and LiNGAM on the same data
• Build intuition for algorithm selection based on assumptions
• Understand consensus edges as most trustworthy

Plots:

• Three DAG outputs side-by-side
  • PC output: CPDAG with mixed directed/undirected edges
  • GES output: Full DAG with directed edges
  • LiNGAM output: Full DAG with edge weight labels
• Below each: summary statistics
  • Edge count, ambiguity count, computation time
  • Edge weights or confidence scores
• Consensus visualization: which edges recovered by each algorithm

Parameters:

• Dataset Type: Nature of data (Linear Gaussian, Linear Non-Gaussian, Nonlinear)
• Sample Size (N): Number of observations (100 to 5000)
• Toggle: show edge weights/confidence scores

Key observations:

• PC: constraint-based, good for small samples, returns equivalence class
• GES: score-based, more directed edges, assumes no hidden confounders
• LiNGAM: functional approach, requires non-Gaussianity, full DAG recovery
• Consensus edges (found by multiple algorithms) are most trustworthy
• Algorithm choice depends on data type, sample size, and assumptions

Cell 8: Validating Discovered DAGs with Refutation Tests¶

Goal:

• Show how to validate that a discovered DAG’s implications hold in the data
• Understand refutation as check on model assumptions
• Build sensitivity to hidden confounding

Plots:

• Left panel: discovered CPDAG from PC or GES
• Right panel: validation dashboard with three sections
  • CI test validation: bar chart of p-values for all implied independencies
    • Green bars: CI holds (p > alpha)
    • Red bars: CI violated (p <= alpha)
  • Placebo test: discovery output when applied to shuffled data
    • Should find nothing or very few edges
  • Sensitivity analysis: line plot showing conclusion robustness vs. hidden confounder strength

Parameters:

• Alpha (CI threshold): Significance level for validation (0.01 to 0.2)
• Confounder Strength: Magnitude of hidden confounding (0 to 1)

Key observations:

• Validation checks whether discovered DAG implications hold in data
• Low validation score suggests violated assumptions (hidden confounders, non-stationarity)
• Placebo tests detect spurious structure discovery in noise
• Sensitivity analysis quantifies robustness to unobserved confounders
• A discovered DAG is a hypothesis; validation checks consistency with data

Cell 9: Domain Knowledge Integration: Constraints and Prior DAGs¶

Goal:

• Show how expert knowledge (forbidden/required edges, temporal order)
• Drastically improves discovery accuracy without losing data-driven insights
• Understand constraint-based discovery as combining human and machine intelligence

Plots:

• Top left: fully automatic discovery result (no constraints)
• Top right: expert-specified constraints listed
  • Forbidden edges (e.g., “outcome cannot cause treatment”)
  • Required edges (e.g., “treatment causes outcome”)
  • Temporal tiers showing variable layers
• Bottom left: discovery with constraints applied
  • Reduced edge candidate space
  • Fewer ambiguities in final DAG
• Bottom right: impact summary
  • Search space reduction percentage
  • Edges affected by constraints

Parameters:

• Prior Strength: How much to trust expert knowledge (0 to 1)

Key observations:

• Partial domain knowledge drastically reduces search space and improves accuracy
• Constraints must be correct: wrong priors inject errors
• Temporal ordering is the strongest constraint
• Expert and automated discovery are complementary
• Combine automatic discovery with domain expertise for best results

Cell 10: End-to-End Workflow: From Data to Validated DAG¶

Goal:

• Integrate all techniques into a complete discovery pipeline
• Show the practical workflow from raw data to validated causal structure
• Understand causal discovery as an iterative, multi-stage process

Plots:

• Multi-panel workflow visualization with six stages:
  1. Data preparation: variable histograms, non-Gaussianity test results, temporal order
  2. Algorithm selection and execution: run selected algorithms with results
  3. Consensus step: Venn diagram of edges found by each algorithm
  4. Refinement step: expert review with constraint options
  5. Validation step: refutation tests with pass/fail results
  6. Final DAG: refined and validated causal structure
• Progress bar showing current workflow stage
• Summary statistics at each stage

Parameters:

• Dataset: Type of data (Synthetic: Linear Gaussian, Linear Non-Gaussian, Nonlinear; Real: economic, health)
• Algorithms: Select which algorithms to run (PC, GES, LiNGAM checkboxes)
• Progress Stage: Current stage in the workflow (0 to 5)

Key observations:

• Discovery is a pipeline, not a single algorithm call
• Multiple algorithms provide robustness via consensus
• Expert review and constraints improve accuracy without losing data-driven insights
• Validation ensures the discovered structure is plausible
• Causal discovery is most powerful when combined with domain knowledge and careful interpretation

Summary¶

Causal discovery is a powerful tool for inferring causal structure from observational data, but it requires careful attention to:

• Assumptions: Different algorithms make different assumptions about the data
• Identifiability: Some structures are inherently unidentifiable from observational data
• Validation: Discovered DAGs must be validated and sensitivity-tested
• Domain Knowledge: Expert knowledge dramatically improves results
• Interpretation: Discovered structures are hypotheses for investigation, not definitive proofs

The workflow balance automated discovery with expert judgment, validated with refutation tests, and interpreted as a hypothesis for investigation.