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THEORY-1137: When an abstraction map between two causal models is learned by optimal transport from paired interventional distributions, a transport cost that rewards pairs of states compatible with the same paired interventions gives much lower held-out abstraction error than a Hamming cost when the intervention set is rich, and higher error when it holds only a few, because with few interventions the compatibility cost is nearly uniform

2 min read

Properties3
StatusProposed
Tagscausality
SourceLIT-1806 — Causal Optimal Transport of Abstractions

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Source

Felekis et al. (2024), LIT-1806, §4.2 (pp. 6–7), §6 (pp. 8–10) and Appendices C and E (pp. 17, 19–23) of arXiv 2312.08107v1; read in NOTE-1342.

What was actually shown

The cost. c_ω(x, x′) = |I| − Σ_ι 1[Cmp(x, ι) ∧ Cmp(x′, ω(ι))]: a base state and an abstracted state are cheaper to match the more paired interventions they are both compatible with (eq. 11, p. 6). “By construction, cω will assign maximum values (|I|) for samples for which it has no available information from the ω map” (p. 17).

The comparisons. Abstraction error of the learned map on a held-out intervention pair, by JSD and Wasserstein distance, 50 repetitions (p. 9).

  • STC with interventions on the parentless smoking variable: the ω-cost beats Hamming in every method, e.g. e_JSD 0.010 against 0.087 for the best COTA setting and 0.013 against 0.093 for independent plans (Table 1, p. 10).
  • LUCAS: the ω-cost is at least as good, by small margins (Table 4, p. 11; Tables 10–11).
  • STC with I = {do(T = 0), do(T = 1)}: Hamming better (0.229 against 0.249, Table 2, p. 10).
  • EBM, the battery data: Hamming better (0.220 against 0.379, Table 3, p. 10).

The authors’ account: a small intervention set makes the ω-cost “almost-uniform and uninformative” (p. 9).

What could have failed. The ω-cost could have won or lost regardless of the size of I.

What this does not say

  • That the gain comes from causal constraints between plans. The do-calculus term is constant on the feasible set (THEORY-1123); the cost is what differs between the compared settings, and the cost acts the same way in the baselines with no causal term.
  • How rich is rich enough. Four cases, with sizes not varied systematically.
  • Anything for continuous variables or large domains. All domains are small and discrete.
  • That the learned maps are causal abstractions in a strong sense. The target is an approximate one-distribution exact transformation, with a stochastic τ (THEORY-849, CLAIM-399).

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