Causal Strength Models #
@cite{icard-et-al-2017}
Theory-layer definitions for causal strength: computational models that rank candidate causes by importance, given a causal model.
Two models are formalized:
Necessity-Sufficiency Model (NSM, @cite{icard-et-al-2017}):
NSM(C) = P(C) · Suf(C) + (1 − P(C)) · Nec(C)Sampling propensity (shared by NSM and CESM):
SP(V) = s · ⟦V⟧^w@ + (1 − s) · P(V)
These definitions are theory-layer infrastructure imported by study files
(e.g., KonukEtAl2026, BellerGerstenberg2025).
Sampling Propensity #
The probability that a variable retains its actual-world value in a counterfactual sample. Interpolates between the actual value (stability parameter s) and the prior probability (1 − s).
Sampling propensity of a binary variable V.
s: stability parameter ∈ [0,1] — probability of copying the actual valueactualValue: ⟦V⟧^w@ ∈ {0,1} — the variable's value in the actual worldpriorProb: P(V) — the prior probability of V being true
Equations
- Semantics.Causation.Strength.samplingPropensity s actualValue priorProb = s * actualValue + (1 - s) * priorProb
Instances For
At s = 0, sampling propensity reduces to the prior.
At s = 1, sampling propensity reduces to the actual value.
Necessity-Sufficiency Model (NSM) #
The causal strength of a candidate cause C for outcome E, computed as a weighted combination of sufficiency and necessity scores.
NSM formula: NSM(C) = P(C) · Suf(C) + (1 − P(C)) · Nec(C).
pC: prior probability (or sampling propensity) of the causesuf: sufficiency score ∈ [0,1]nec: necessity score ∈ [0,1]
Equations
- Semantics.Causation.Strength.nsm pC suf nec = pC * suf + (1 - pC) * nec
Instances For
When Suf = 1, NSM simplifies to P(C) + (1 − P(C)) · Nec.
Sufficient and necessary: NSM = 1 regardless of probability.
Sufficient but not necessary: NSM = P(C).
Necessary but not sufficient: NSM = 1 − P(C).
Neither sufficient nor necessary: NSM = 0.