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Causal Discovery: Part 6: Evaluation and ML Applications to References

6. Evaluation and ML Applications

Evaluation and ML Applications develops the part of causal discovery specified by the approved Chapter 22 table of contents. The treatment is causal, not merely predictive: the central objects are mechanisms, interventions, assumptions, and counterfactuals.

6.1 structural Hamming distance

Structural hamming distance belongs to the canonical scope of Causal Discovery. The central move in causal inference is to distinguish a statistical relation from a claim about what would happen under an intervention.

For this subsection, the working scope is constraint-based, score-based, functional, invariant, and optimization-based causal graph discovery with clear assumptions and evaluation metrics. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$h(W)=\operatorname{tr}(e^{W\odot W})-d=0.$$

The formula gives a compact handle on structural hamming distance. It should not be read as a purely algebraic identity. In causal inference, equations encode assumptions about mechanisms, missing variables, and which parts of the world remain stable under intervention.

Three examples of structural hamming distance:

1. A recommender team wants the causal effect of ranking a document higher, not merely the correlation between rank and clicks.
2. An LLM platform changes a safety policy and wants to estimate whether refusals changed because of the policy or because user prompts shifted.
3. A fairness auditor asks whether a proxy feature transmits an impermissible causal path into a model decision.

Two non-examples expose the boundary:

1. A high predictive coefficient is not a causal effect unless the graph and intervention assumptions justify it.
2. A plausible narrative produced by a language model is not a counterfactual unless it is grounded in a causal model.

The proof habit for structural hamming distance is to name the graph operation. Conditioning restricts a distribution. Intervention replaces a mechanism. Counterfactual reasoning updates exogenous uncertainty from evidence, changes a mechanism, then predicts.

observed association:      P(Y | X=x)
intervention question:     P(Y | do(X=x))
counterfactual question:   P(Y_x | E=e)
discovery question:        which G could have generated P(V)?

In machine learning, structural hamming distance is valuable because models are often deployed under interventions: ranking changes, policy changes, safety filters, tool-use gates, data collection changes, and human feedback loops. Prediction alone does not tell us which change caused which downstream behavior.

Notebook implementation will use synthetic SCMs and small graphs. This keeps the examples executable while preserving the conceptual split between identification and estimation.

Checklist for using structural hamming distance responsibly:

• State the causal question before choosing a method.
• Draw or describe the assumed causal graph.
• Mark observed, latent, treatment, outcome, and adjustment variables.
• Separate intervention notation from conditioning notation.
• Decide whether the query is identifiable before estimating it.
• Report assumptions that cannot be tested from the observed data alone.
• Use ML as an estimation aid, not as a substitute for causal design.

This chapter follows the boundary set by Chapter 21. Statistical learning theory controls prediction error under distributional assumptions. Causal inference asks what happens when the distribution changes because something is done.

Modern AI systems make this distinction unavoidable. A foundation model can predict which action historically followed a context, but a decision system needs to know what would happen if it took a different action in that context.

Thus, structural hamming distance is not an abstract philosophical add-on. It is a production and research tool for deciding which model, prompt, policy, feature, or intervention actually changed an outcome.

A final diagnostic question is whether the claim would survive a policy change. If the answer depends only on a historical correlation, it belongs in predictive modeling. If the answer depends on what mechanism is replaced and which paths remain active, it belongs in causal inference.

6.2 structural intervention distance preview

Structural intervention distance preview belongs to the canonical scope of Causal Discovery. The central move in causal inference is to distinguish a statistical relation from a claim about what would happen under an intervention.

For this subsection, the working scope is constraint-based, score-based, functional, invariant, and optimization-based causal graph discovery with clear assumptions and evaluation metrics. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$\operatorname{SHD}(G,\widehat{G})=\#\{\text{edge additions, deletions, reversals}\}.$$

The formula gives a compact handle on structural intervention distance preview. It should not be read as a purely algebraic identity. In causal inference, equations encode assumptions about mechanisms, missing variables, and which parts of the world remain stable under intervention.

Three examples of structural intervention distance preview:

1. A recommender team wants the causal effect of ranking a document higher, not merely the correlation between rank and clicks.
2. An LLM platform changes a safety policy and wants to estimate whether refusals changed because of the policy or because user prompts shifted.
3. A fairness auditor asks whether a proxy feature transmits an impermissible causal path into a model decision.

Two non-examples expose the boundary:

1. A high predictive coefficient is not a causal effect unless the graph and intervention assumptions justify it.
2. A plausible narrative produced by a language model is not a counterfactual unless it is grounded in a causal model.

The proof habit for structural intervention distance preview is to name the graph operation. Conditioning restricts a distribution. Intervention replaces a mechanism. Counterfactual reasoning updates exogenous uncertainty from evidence, changes a mechanism, then predicts.

observed association:      P(Y | X=x)
intervention question:     P(Y | do(X=x))
counterfactual question:   P(Y_x | E=e)
discovery question:        which G could have generated P(V)?

In machine learning, structural intervention distance preview is valuable because models are often deployed under interventions: ranking changes, policy changes, safety filters, tool-use gates, data collection changes, and human feedback loops. Prediction alone does not tell us which change caused which downstream behavior.

Notebook implementation will use synthetic SCMs and small graphs. This keeps the examples executable while preserving the conceptual split between identification and estimation.

Checklist for using structural intervention distance preview responsibly:

• State the causal question before choosing a method.
• Draw or describe the assumed causal graph.
• Mark observed, latent, treatment, outcome, and adjustment variables.
• Separate intervention notation from conditioning notation.
• Decide whether the query is identifiable before estimating it.
• Report assumptions that cannot be tested from the observed data alone.
• Use ML as an estimation aid, not as a substitute for causal design.

This chapter follows the boundary set by Chapter 21. Statistical learning theory controls prediction error under distributional assumptions. Causal inference asks what happens when the distribution changes because something is done.

Modern AI systems make this distinction unavoidable. A foundation model can predict which action historically followed a context, but a decision system needs to know what would happen if it took a different action in that context.

Thus, structural intervention distance preview is not an abstract philosophical add-on. It is a production and research tool for deciding which model, prompt, policy, feature, or intervention actually changed an outcome.

A final diagnostic question is whether the claim would survive a policy change. If the answer depends only on a historical correlation, it belongs in predictive modeling. If the answer depends on what mechanism is replaced and which paths remain active, it belongs in causal inference.

6.3 synthetic benchmarks

Synthetic benchmarks belongs to the canonical scope of Causal Discovery. The central move in causal inference is to distinguish a statistical relation from a claim about what would happen under an intervention.

For this subsection, the working scope is constraint-based, score-based, functional, invariant, and optimization-based causal graph discovery with clear assumptions and evaluation metrics. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$X_j=f_j(\operatorname{pa}_j)+N_j,\qquad N_j \perp\!\!\!\perp \operatorname{pa}_j.$$

The formula gives a compact handle on synthetic benchmarks. It should not be read as a purely algebraic identity. In causal inference, equations encode assumptions about mechanisms, missing variables, and which parts of the world remain stable under intervention.

Three examples of synthetic benchmarks:

1. A recommender team wants the causal effect of ranking a document higher, not merely the correlation between rank and clicks.
2. An LLM platform changes a safety policy and wants to estimate whether refusals changed because of the policy or because user prompts shifted.
3. A fairness auditor asks whether a proxy feature transmits an impermissible causal path into a model decision.

Two non-examples expose the boundary:

1. A high predictive coefficient is not a causal effect unless the graph and intervention assumptions justify it.
2. A plausible narrative produced by a language model is not a counterfactual unless it is grounded in a causal model.

The proof habit for synthetic benchmarks is to name the graph operation. Conditioning restricts a distribution. Intervention replaces a mechanism. Counterfactual reasoning updates exogenous uncertainty from evidence, changes a mechanism, then predicts.

observed association:      P(Y | X=x)
intervention question:     P(Y | do(X=x))
counterfactual question:   P(Y_x | E=e)
discovery question:        which G could have generated P(V)?

In machine learning, synthetic benchmarks is valuable because models are often deployed under interventions: ranking changes, policy changes, safety filters, tool-use gates, data collection changes, and human feedback loops. Prediction alone does not tell us which change caused which downstream behavior.

Notebook implementation will use synthetic SCMs and small graphs. This keeps the examples executable while preserving the conceptual split between identification and estimation.

Checklist for using synthetic benchmarks responsibly:

• State the causal question before choosing a method.
• Draw or describe the assumed causal graph.
• Mark observed, latent, treatment, outcome, and adjustment variables.
• Separate intervention notation from conditioning notation.
• Decide whether the query is identifiable before estimating it.
• Report assumptions that cannot be tested from the observed data alone.
• Use ML as an estimation aid, not as a substitute for causal design.

This chapter follows the boundary set by Chapter 21. Statistical learning theory controls prediction error under distributional assumptions. Causal inference asks what happens when the distribution changes because something is done.

Modern AI systems make this distinction unavoidable. A foundation model can predict which action historically followed a context, but a decision system needs to know what would happen if it took a different action in that context.

Thus, synthetic benchmarks is not an abstract philosophical add-on. It is a production and research tool for deciding which model, prompt, policy, feature, or intervention actually changed an outcome.

A final diagnostic question is whether the claim would survive a policy change. If the answer depends only on a historical correlation, it belongs in predictive modeling. If the answer depends on what mechanism is replaced and which paths remain active, it belongs in causal inference.

6.4 causal feature selection

Causal feature selection belongs to the canonical scope of Causal Discovery. The central move in causal inference is to distinguish a statistical relation from a claim about what would happen under an intervention.

For this subsection, the working scope is constraint-based, score-based, functional, invariant, and optimization-based causal graph discovery with clear assumptions and evaluation metrics. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$A_{ij}=1 \iff X_i \to X_j.$$

The formula gives a compact handle on causal feature selection. It should not be read as a purely algebraic identity. In causal inference, equations encode assumptions about mechanisms, missing variables, and which parts of the world remain stable under intervention.

Three examples of causal feature selection:

1. A recommender team wants the causal effect of ranking a document higher, not merely the correlation between rank and clicks.
2. An LLM platform changes a safety policy and wants to estimate whether refusals changed because of the policy or because user prompts shifted.
3. A fairness auditor asks whether a proxy feature transmits an impermissible causal path into a model decision.

Two non-examples expose the boundary:

1. A high predictive coefficient is not a causal effect unless the graph and intervention assumptions justify it.
2. A plausible narrative produced by a language model is not a counterfactual unless it is grounded in a causal model.

The proof habit for causal feature selection is to name the graph operation. Conditioning restricts a distribution. Intervention replaces a mechanism. Counterfactual reasoning updates exogenous uncertainty from evidence, changes a mechanism, then predicts.

observed association:      P(Y | X=x)
intervention question:     P(Y | do(X=x))
counterfactual question:   P(Y_x | E=e)
discovery question:        which G could have generated P(V)?

In machine learning, causal feature selection is valuable because models are often deployed under interventions: ranking changes, policy changes, safety filters, tool-use gates, data collection changes, and human feedback loops. Prediction alone does not tell us which change caused which downstream behavior.

Notebook implementation will use synthetic SCMs and small graphs. This keeps the examples executable while preserving the conceptual split between identification and estimation.

Checklist for using causal feature selection responsibly:

• State the causal question before choosing a method.
• Draw or describe the assumed causal graph.
• Mark observed, latent, treatment, outcome, and adjustment variables.
• Separate intervention notation from conditioning notation.
• Decide whether the query is identifiable before estimating it.
• Report assumptions that cannot be tested from the observed data alone.
• Use ML as an estimation aid, not as a substitute for causal design.

This chapter follows the boundary set by Chapter 21. Statistical learning theory controls prediction error under distributional assumptions. Causal inference asks what happens when the distribution changes because something is done.

Modern AI systems make this distinction unavoidable. A foundation model can predict which action historically followed a context, but a decision system needs to know what would happen if it took a different action in that context.

Thus, causal feature selection is not an abstract philosophical add-on. It is a production and research tool for deciding which model, prompt, policy, feature, or intervention actually changed an outcome.

A final diagnostic question is whether the claim would survive a policy change. If the answer depends only on a historical correlation, it belongs in predictive modeling. If the answer depends on what mechanism is replaced and which paths remain active, it belongs in causal inference.

6.5 LLM-assisted causal hypothesis generation with human review

Llm-assisted causal hypothesis generation with human review belongs to the canonical scope of Causal Discovery. The central move in causal inference is to distinguish a statistical relation from a claim about what would happen under an intervention.

For this subsection, the working scope is constraint-based, score-based, functional, invariant, and optimization-based causal graph discovery with clear assumptions and evaluation metrics. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$h(W)=\operatorname{tr}(e^{W\odot W})-d=0.$$

The formula gives a compact handle on llm-assisted causal hypothesis generation with human review. It should not be read as a purely algebraic identity. In causal inference, equations encode assumptions about mechanisms, missing variables, and which parts of the world remain stable under intervention.

Three examples of llm-assisted causal hypothesis generation with human review:

1. A recommender team wants the causal effect of ranking a document higher, not merely the correlation between rank and clicks.
2. An LLM platform changes a safety policy and wants to estimate whether refusals changed because of the policy or because user prompts shifted.
3. A fairness auditor asks whether a proxy feature transmits an impermissible causal path into a model decision.

Two non-examples expose the boundary:

1. A high predictive coefficient is not a causal effect unless the graph and intervention assumptions justify it.
2. A plausible narrative produced by a language model is not a counterfactual unless it is grounded in a causal model.

The proof habit for llm-assisted causal hypothesis generation with human review is to name the graph operation. Conditioning restricts a distribution. Intervention replaces a mechanism. Counterfactual reasoning updates exogenous uncertainty from evidence, changes a mechanism, then predicts.

observed association:      P(Y | X=x)
intervention question:     P(Y | do(X=x))
counterfactual question:   P(Y_x | E=e)
discovery question:        which G could have generated P(V)?

In machine learning, llm-assisted causal hypothesis generation with human review is valuable because models are often deployed under interventions: ranking changes, policy changes, safety filters, tool-use gates, data collection changes, and human feedback loops. Prediction alone does not tell us which change caused which downstream behavior.

Notebook implementation will use synthetic SCMs and small graphs. This keeps the examples executable while preserving the conceptual split between identification and estimation.

Checklist for using llm-assisted causal hypothesis generation with human review responsibly:

• State the causal question before choosing a method.
• Draw or describe the assumed causal graph.
• Mark observed, latent, treatment, outcome, and adjustment variables.
• Separate intervention notation from conditioning notation.
• Decide whether the query is identifiable before estimating it.
• Report assumptions that cannot be tested from the observed data alone.
• Use ML as an estimation aid, not as a substitute for causal design.

This chapter follows the boundary set by Chapter 21. Statistical learning theory controls prediction error under distributional assumptions. Causal inference asks what happens when the distribution changes because something is done.

Modern AI systems make this distinction unavoidable. A foundation model can predict which action historically followed a context, but a decision system needs to know what would happen if it took a different action in that context.

Thus, llm-assisted causal hypothesis generation with human review is not an abstract philosophical add-on. It is a production and research tool for deciding which model, prompt, policy, feature, or intervention actually changed an outcome.

A final diagnostic question is whether the claim would survive a policy change. If the answer depends only on a historical correlation, it belongs in predictive modeling. If the answer depends on what mechanism is replaced and which paths remain active, it belongs in causal inference.

7. Common Mistakes

8. Exercises

1. (*) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

2. (*) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

3. (*) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

4. (**) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

5. (**) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

6. (**) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

7. (***) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

8. (***) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

9. (***) Work through a causal-inference task for causal discovery.

   • (a) State the causal query using intervention or counterfactual notation.
   • (b) Draw or describe the relevant graph and assumptions.
   • (c) Decide whether the estimand is identifiable from the available data.
   • (d) Give an estimator or diagnostic only after identification is clear.
   • (e) Explain the AI or LLM system implication.

10. (***) Work through a causal-inference task for causal discovery.

• (a) State the causal query using intervention or counterfactual notation.
• (b) Draw or describe the relevant graph and assumptions.
• (c) Decide whether the estimand is identifiable from the available data.
• (d) Give an estimator or diagnostic only after identification is clear.
• (e) Explain the AI or LLM system implication.

9. Why This Matters for AI

10. Conceptual Bridge

Causal Discovery follows statistical learning theory because learning theory explains how observed samples support future prediction claims. Causal inference asks a different question: what happens when an action changes the system that generated those samples?

The backward bridge is risk and uncertainty. Chapter 21 provides language for finite-sample generalization. Chapter 22 adds intervention semantics, graph assumptions, and counterfactual worlds. A causal claim is not just a better prediction; it is a claim about a modified data-generating mechanism.

The forward bridge is game theory. Once multiple agents adapt to interventions, the causal question becomes strategic: actions change incentives, incentives change behavior, and behavior changes the causal system. Chapter 23 will study that interaction explicitly.

+--------------------------------------------------------------+
| Chapter 21: prediction under finite samples                  |
| Chapter 22: intervention, counterfactuals, causal discovery  |
| Chapter 23: strategic interaction and adversarial systems    |
+--------------------------------------------------------------+

References

• Spirtes, Glymour, and Scheines. Causation, Prediction, and Search. https://mitpress.mit.edu/9780262194402/causation-prediction-and-search/
• Peters, Janzing, and Scholkopf. Elements of Causal Inference. https://tile.loc.gov/storage-services/master/gdc/gdcebookspublic/20/20/71/97/58/2020719758/2020719758.pdf
• Zheng et al.. DAGs with NO TEARS: Continuous Optimization for Structure Learning. https://arxiv.org/abs/1803.01422
• Peters et al.. Causal discovery with continuous additive noise models. https://jmlr.org/papers/v15/peters14a.html