Deterministic Model Parameters
When a structural model is deterministic (no background noise), the probabilistic parameters are:
- P(C|A) = 1 if A is sufficient for C
- P(C|A) = 0 if A is not sufficient for C
- P(C|¬A) = 0 if A is necessary for C
- P(C|¬A) = 1 if C occurs without A
- sufficient : Bool
Whether cause is sufficient for effect
- necessary : Bool
Whether cause is necessary for effect
- alternativeCause : Bool
Whether effect can occur without cause (alternative cause present)
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Convert deterministic parameters to conditional probabilities.
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Convert deterministic parameters to NoisyOR parameters.
In a deterministic model:
- background = P(C|¬A) = 1 if alternative cause, else 0
- power = P(C|A) - P(C|¬A)
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Build DeterministicParams from a CausalProfile.
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- NadathurLauer2020.Integration.DeterministicParams.ofProfile p = { sufficient := p.sufficient, necessary := p.necessary, alternativeCause := !p.necessary }
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Extract deterministic parameters from a structural causal model.
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Situation to WorldState: Convert a structural situation to a probabilistic world state.
Given:
- A structural causal model (dynamics + situation)
- Prior probability P(cause) for the cause variable
Compute:
- WorldState with P(A), P(C), P(A∧C)
Assumptions:
- The dynamics are deterministic (each law fires with probability 1)
- We're computing the induced distribution given the causal structure
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Alternative: create WorldState assuming uniform prior P(cause) = 1/2.
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- NadathurLauer2020.Integration.situationToWorldStateUniform dyn background cause effect = NadathurLauer2020.Integration.situationToWorldState dyn background cause effect (1 / 2)
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Sufficiency implies high P(C|A).
If A is structurally sufficient for C, then P(C|A) = 1 in the induced probabilistic model.
Necessity implies P(C|¬A) = 0 (when no alternative cause).
If A is structurally necessary for C (and no alternative causes), then P(C|¬A) = 0 in the induced probabilistic model.
Structural causation grounds pragmatic inference.
When the underlying causal model has A both sufficient AND necessary for C, the RSA listener should infer CausalRelation.ACausesC.
This connects:
- Semantic truth conditions (Nadathur & Lauer)
- Pragmatic inference (Grusdt et al.)
Map structural causation to CausalRelation.
| Sufficient | Necessary | CausalRelation |
|---|---|---|
| true | true | ACausesC |
| true | false | Independent (overdetermination) |
| false | true | (partial cause) |
| false | false | Independent |
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Extract CausalRelation directly from a structural model.
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The Grounding Chain
This theorem shows how structural causation grounds probabilistic inference:
CausalDynamics (structural model)
↓ extractParams
DeterministicParams (sufficient? necessary?)
↓ toConditionals
P(C|A), P(C|¬A) (conditional probabilities)
↓ situationToWorldState
WorldState (full probability distribution)
↓ inferCausalRelation
CausalRelation (pragmatic inference)
When the structural model has A→C as sufficient and necessary, the entire chain produces ACausesC.
Overdetermination creates a disconnect.
In overdetermination:
- Structurally: A is sufficient but NOT necessary
- Probabilistically: High P(C|A), but also high P(C|¬A)
- Inference: Not clearly ACausesC (could be common cause, or multiple causes)
This shows why "cause" is false but "make" is true in overdetermination.
Causal Perfection: The pragmatic inference from sufficiency to necessity.
When a speaker asserts "X made Y happen" (sufficiency):
- Why didn't they just say "Y happened"?
- Gricean inference: X must be important (not just sufficient, but necessary)
This connects to conditional perfection from Grusdt et al.:
- "If A then C" → pragmatic inference → "If not-A then not-C"
- "X made Y" → pragmatic inference → "X caused Y"
The structural grounding makes this inference reasonable:
- If A were not necessary, there would be alternative causes
- A rational speaker would mention those alternatives
- Silence about alternatives → A was probably necessary too
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- NadathurLauer2020.Integration.causalPerfectionInference dyn background cause effect = Core.StructuralEquationModel.causallySufficient dyn background cause effect
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When there's a single cause in the model, sufficiency implies necessity. Under @cite{nadathur-2024} Def 10b, necessity is tested with the cause NOT in the background (the precondition requires cause not already entailed).