Lesson overview | Previous part | Next part

Structural Causal Models: Part 2: Formal Definitions

2. Formal Definitions

Formal Definitions develops the part of structural causal models 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.

2.1 exogenous variables $\mathbf{U}$

Exogenous variables $\mathbf{u}$ belongs to the canonical scope of Structural Causal Models. 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 structural assignments, causal graphs, d-separation, interventions, Markovian assumptions, and SCM links to robust ML. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$V_i=f_i(\operatorname{pa}_i,U_i).$$

The formula gives a compact handle on exogenous variables $\mathbf{u}$. 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 exogenous variables $\mathbf{u}$:

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 exogenous variables $\mathbf{u}$ 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, exogenous variables $\mathbf{u}$ 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 exogenous variables $\mathbf{u}$ 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, exogenous variables $\mathbf{u}$ 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.

2.2 endogenous variables $\mathbf{V}$

Endogenous variables $\mathbf{v}$ belongs to the canonical scope of Structural Causal Models. 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 structural assignments, causal graphs, d-separation, interventions, Markovian assumptions, and SCM links to robust ML. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$P(\mathbf{v})=\prod_{i=1}^{d}P(v_i \mid \operatorname{pa}_i).$$

The formula gives a compact handle on endogenous variables $\mathbf{v}$. 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 endogenous variables $\mathbf{v}$:

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 endogenous variables $\mathbf{v}$ 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, endogenous variables $\mathbf{v}$ 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 endogenous variables $\mathbf{v}$ 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, endogenous variables $\mathbf{v}$ 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.

2.3 structural assignments $V_i=f_i(\operatorname{pa}_i,U_i)$

Structural assignments $v_i=f_i(\operatorname{pa}_i,u_i)$ belongs to the canonical scope of Structural Causal Models. 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 structural assignments, causal graphs, d-separation, interventions, Markovian assumptions, and SCM links to robust ML. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$P_M(Y \mid \operatorname{do}(X=x))=P_{M_x}(Y).$$

The formula gives a compact handle on structural assignments $v_i=f_i(\operatorname{pa}_i,u_i)$. 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 assignments $v_i=f_i(\operatorname{pa}_i,u_i)$:

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 assignments $v_i=f_i(\operatorname{pa}_i,u_i)$ 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 assignments $v_i=f_i(\operatorname{pa}_i,u_i)$ 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 assignments $v_i=f_i(\operatorname{pa}_i,u_i)$ 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 assignments $v_i=f_i(\operatorname{pa}_i,u_i)$ 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.

2.4 causal graph $G$

Causal graph $g$ belongs to the canonical scope of Structural Causal Models. 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 structural assignments, causal graphs, d-separation, interventions, Markovian assumptions, and SCM links to robust ML. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$M=(\mathbf{U},\mathbf{V},\mathbf{F},P(\mathbf{U})).$$

The formula gives a compact handle on causal graph $g$. 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 graph $g$:

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 graph $g$ 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 graph $g$ 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 graph $g$ 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 graph $g$ 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.

2.5 observational interventional and counterfactual distributions

Observational interventional and counterfactual distributions belongs to the canonical scope of Structural Causal Models. 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 structural assignments, causal graphs, d-separation, interventions, Markovian assumptions, and SCM links to robust ML. The mathematical objects are variables, mechanisms, graphs, interventions, and assumptions. A causal claim is incomplete until all five are visible.

$$V_i=f_i(\operatorname{pa}_i,U_i).$$

The formula gives a compact handle on observational interventional and counterfactual distributions. 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 observational interventional and counterfactual distributions:

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 observational interventional and counterfactual distributions 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, observational interventional and counterfactual distributions 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 observational interventional and counterfactual distributions 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, observational interventional and counterfactual distributions 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.