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.

Imports¶

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

Cell 1.1: Two DAGs, one correlation¶

Goal:

• Recognize that identical observational patterns can hide opposite causal implications

Implementation: utils.cell1_correlation_vs_causation()

• Generates data from a chain X -> Y and a reverse Z -> Y -> X structure that produce the same X/Y correlation

Usage

• Inputs

  • seed: random seed for the sampled data
  • Correlation (r): strength of correlation between $X$ and $Y$
  • Mode: toggle between observational and interventional framing

• Panels

  • Chain: X -> Y: scatter plot, with a note that intervening on $X$ changes $Y$
  • Reverse: Z -> Y -> X: scatter plot with the identical correlation, but intervening on $X$ has no effect
  • Comments: current seed and correlation, plus the key takeaway

Guided usage

• Raise Correlation (r) toward its max
  • Observe both panels’ correlation grows identically, even though one structure is causal and the other is not
• Compare the 2 panels at any setting
  • Observe prediction and causation require fundamentally different reasoning: no amount of observational data alone distinguishes these structures, we need additional assumptions or interventions

Part 2: Markov Equivalence: Three Indistinguishable Structures¶

Cell 2.1: Same independencies, different DAGs¶

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

Implementation: utils.cell2_markov_equivalence()

• Generates data from a chain X -> Y -> Z, then computes the marginal and conditional (given $Y$) correlation between $X$ and $Z$

Usage

• Inputs

  • seed: random seed for the sampled data
  • Sample Size (N): number of observations

• Panels

  • Chain/Reverse/Common Cause: the 3 Markov-equivalent DAGs
  • CI text: states the shared independence $X \perp Z \mid Y$
  • Marginal Correlation/Conditional Correlation: bar charts of corr($X$, $Z$) and corr($X$, $Z \mid Y$)
  • Comments: current seed, $N$, and the 2 correlation values

Guided usage

• Raise Sample Size (N) from its min to its max
  • Observe the marginal and conditional correlations stay about the same: larger samples provide stronger statistical power for CI tests, but even infinite samples cannot distinguish Markov equivalent structures
• Compare the 3 DAGs
  • Observe all 3 satisfy the same independencies and have identical covariance matrices, so observational data alone recovers only the CPDAG, never a unique DAG

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

Cell 3.1: Same correlation, different interventions¶

Goal:

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

Implementation: utils.cell3_causal_effects()

• Simulates intervening on $X$ (set to a high value) and measures the effect on $Z$ under 3 structures: chain, reverse, common cause

Usage

• Inputs

  • seed: random seed for the sampled data
  • Intervention Strength: magnitude of the change applied to $X$
  • Sample Size (N): observations used for effect estimation

• Panels

  • Chain: X -> Y -> Z: large effect on $Z$ (green bar)
  • Reverse: Z -> Y -> X: no effect on $Z$ (red bar)
  • Common Cause: no direct effect on $Z$ (orange bar)
  • Comments: current parameters and the key takeaway

Guided usage

• Raise Intervention Strength
  • Observe only the chain’s bar grows: the reverse and common-cause bars stay at 0 regardless of intervention strength
• Compare the 3 bars at any setting
  • Observe the same observational correlation produces wildly different causal effects: choosing the wrong DAG leads to ineffective or harmful interventions

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

Cell 4.1: Testing independence, then orienting edges¶

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

Implementation: utils.cell4_pc_algorithm()

• Steps through a simplified PC run on a 3-node graph: test, remove edges that pass, then orient v-structures

Usage

• Inputs

  • Alpha (CI threshold): significance level for CI tests
  • Test Type: CI test method (partial correlation, Gaussian G-squared, conditional MI)
  • Speed: animation speed for the step-by-step progression

• Panels

  • PC Algorithm Progression: the evolving graph structure as edges are removed
  • Current Test: the current step, its p-value, and the separating set
  • Comments: current parameters and the algorithm’s guarantees

Guided usage

• Lower Alpha (CI threshold)
  • Observe stricter tests keep more edges
• Raise Alpha (CI threshold)
  • Observe more liberal tests remove more edges
• Switch Test Type
  • Observe stronger distributional assumptions enable more powerful tests; PC is sound in the large-sample limit, but CI tests are underpowered in finite samples

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

Cell 5.1: Optimizing a score instead of testing independence¶

Goal:

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

Implementation: utils.cell5_ges_algorithm()

• Simulates a BIC trajectory across search iterations, with a forward/backward phase transition marker

Usage

• Inputs

  • Sample Size (N): number of observations (the BIC penalty is proportional to $\log(N)$)
  • Regularization: strength of the sparsity penalty

• Panels

  • Current DAG Structure: the DAG found at the current settings
  • Score Trajectory (GES Search): BIC score vs. iteration, with the forward/backward transition marked
  • Comments: current parameters and the search strategy

Guided usage

• Raise Sample Size (N)
  • Observe larger $N$ allows more edges, since the BIC complexity penalty ($\propto \log N$) grows slower than the likelihood gain
• Raise Regularization
  • Observe fewer edges survive; greedy search can get stuck in local optima, so multiple random starts improve robustness

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

Cell 6.1: Breaking symmetry with non-Gaussian noise¶

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

Implementation: utils.cell6_lingam_nongaussian()

• Generates a chain with skewed (or Gaussian) noise, and runs a Jarque-Bera test for non-Gaussianity

Usage

• Inputs

  • seed: random seed for the sampled data
  • Skewness: non-Gaussianity of the noise distribution (0 = Gaussian)
  • Signal-to-Noise Ratio: ratio of signal to noise strength

• Panels

  • X -> Y/Y -> Z/X -> Z (indirect): scatter plots of each pair
  • Comments: current parameters, the Jarque-Bera test result, and the key takeaway

Guided usage

• Set Skewness to 0
  • Observe the Jarque-Bera test in Comments cannot reject normality, and the chains are symmetric, so their direction is indistinguishable
• Raise Skewness
  • Observe the test rejects normality, and the scatter plots become visibly asymmetric: LiNGAM exploits this asymmetry to orient edges and recover the full DAG

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

Cell 7.1: PC, GES, and LiNGAM on the same data¶

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

Implementation: utils.cell7_algorithm_comparison()

Usage

• Inputs

  • Dataset Type: nature of the data (linear Gaussian, linear non-Gaussian, nonlinear)
  • Sample Size (N): number of observations

• Panels

  • PC (Constraint-Based): CPDAG output, with an edge/ambiguity count
  • GES (Score-Based): full DAG output
  • LiNGAM (Functional): full DAG output
  • Comments: current parameters and each algorithm’s tradeoffs

Guided usage

• Switch Dataset Type across all 3 options
  • Observe the edge/ambiguity counts stay fixed in this simplified demo, but in practice: PC is good for exploratory analysis, GES assumes no hidden confounders, and LiNGAM requires non-Gaussianity
• Raise Sample Size (N)
  • Observe the Comments panel updates with the new sample size; consensus edges found by multiple algorithms are the most trustworthy

Part 8: Validating Discovered DAGs with Refutation Tests¶

Cell 8.1: Checking whether a DAG’s implications hold¶

Goal:

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

Implementation: utils.cell8_validation()

Usage

• Inputs

  • Alpha (CI threshold): significance level for validation
  • Confounder Strength: magnitude of hidden confounding

• Panels

  • CI Test Validation: bar chart of p-values for implied independencies (green: holds, red: violated)
  • Placebo Test: discovery output on shuffled data (should find ~0 edges)
  • Sensitivity Analysis: robustness vs. hidden confounder strength
  • Comments: current parameters and the validation score

Guided usage

• Lower Alpha (CI threshold)
  • Observe fewer bars in CI Test Validation turn red, since the threshold for “violated” becomes stricter
• Raise Confounder Strength
  • Observe the Sensitivity Analysis robustness curve drops faster: a discovered DAG is a hypothesis, and validation checks its consistency with the data, it does not prove it

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

Cell 9.1: Combining expert knowledge with discovery¶

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

Implementation: utils.cell9_domain_knowledge()

Usage

• Inputs

  • Prior Strength: how much to trust expert knowledge

• Panels

  • No Constraints: fully automatic discovery result
  • With Constraints: discovery with the expert constraints applied, fewer ambiguities
  • Comments: the expert constraints, and the impact summary (search-space reduction, edges affected)

Guided usage

• Raise Prior Strength
  • Observe the impact summary in Comments reports a bigger search- space reduction
• Compare No Constraints vs With Constraints
  • Observe temporal ordering is the strongest constraint: expert and automated discovery are complementary, and combining them gives the best results, but wrong priors inject errors

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

Cell 10.1: The full discovery pipeline¶

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

Implementation: utils.cell10_end_to_end_workflow()

• 6 stages: data preparation, algorithm selection, consensus, refinement, validation, and the final DAG

Usage

• Inputs

  • Dataset: type of data (synthetic: linear Gaussian, non-Gaussian; real: economic)
  • PC/GES/LiNGAM: checkboxes selecting which algorithms to run
  • Progress Stage: current stage in the workflow (0 to 5)

• Panels

  • Stage 1-6: one panel per pipeline stage, highlighted once reached
  • Overall Progress: a progress bar over all 6 stages

Guided usage

• Move Progress Stage from 0 to 5
  • Observe each stage panel highlights in turn, and the progress bar fills up: discovery is a pipeline, not a single algorithm call
• Toggle the PC/GES/LiNGAM checkboxes
  • Observe the Stage 2: Algorithms panel updates with the selected set; multiple algorithms provide robustness via consensus

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