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Causal Discovery: Part 4: Score-Based Discovery

4. Score-Based Discovery

Score-Based Discovery 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.

4.1 likelihood and BIC scores

Likelihood and bic scores 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 likelihood and bic scores. 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 likelihood and bic scores:

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 likelihood and bic scores 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, likelihood and bic scores 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 likelihood and bic scores 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, likelihood and bic scores 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.

4.2 greedy equivalence search

Greedy equivalence search 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 greedy equivalence search. 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 greedy equivalence search:

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 greedy equivalence search 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, greedy equivalence search 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 greedy equivalence search 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, greedy equivalence search 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.

4.3 super-exponential DAG search

Super-exponential dag search 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 super-exponential dag search. 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 super-exponential dag search:

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 super-exponential dag search 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, super-exponential dag search 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 super-exponential dag search 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, super-exponential dag search 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.

4.4 NOTEARS continuous optimization

Notears continuous optimization 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 notears continuous optimization. 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 notears continuous optimization:

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 notears continuous optimization 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, notears continuous optimization 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 notears continuous optimization 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, notears continuous optimization 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.

4.5 regularization and sparsity

Regularization and sparsity 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 regularization and sparsity. 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 regularization and sparsity:

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 regularization and sparsity 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, regularization and sparsity 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 regularization and sparsity 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, regularization and sparsity 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.