Operationalising Weaker Faithfulness: Recovery, Ambiguity and Computational Limits

Mechanistic interpretability aims to recover the internal computations responsible for model behavior. Progress in automated circuit discovery is often framed as a search problem: better attribution or optimization should identify better mechanisms. This assumes that the evaluation objective can recognize a better circuit once it is found. Yet standard intervention-based evaluation replaces signals outside a circuit to isolate its effect, changing the computational environment in which the retained components operate. We call this change context distortion. This raises the question of whether faithfulness rewards recovery of model behavior or compatibility with the altered context. We show that intervention-based faithfulness can prefer an equally sized circuit that reproduces the model's behavior less well. We call this reversal misranking, an objective-level recovery gap. To isolate evaluation from discovery, we construct equally sized candidates through controlled edits to human-identified circuits in pretrained language models, including IOI and Greater-Than, and benchmark circuits with known mechanisms. We find misranking even without a discovery algorithm. Misranking also occurs in real automated circuit-discovery settings. Among outputs of EAP, EAP-IG, ACDC, and Edge-SP, KL faithfulness under ordinary resampling prefers the behaviorally worse candidate in 13.2%-24.4% of candidate pairs across these methods on four pretrained-language-model tasks. Finally, restoring selected intact-model signals reverses 96 of 100 selected persistent misrankings, demonstrating that context distortion contributes to these failures. In light of these findings, we call for rethinking both how circuit discovery is performed and how its results are evaluated.

Causal discovery algorithms learn a network that describes the causal dependencies among random variables. A common workflow involves first utilizing conditional independence properties on observational data to determine partially directed causal relationships, then applying interventions to orient the unknown causal directions. A critical assumption for the first step is faithfulness: a requirement that causally linked variables exhibit statistical dependence. Many natural systems include buffering and stabilizing pathways that cancel out to achieve systemic robustness. This cancellation of pathways violates faithfulness, leading causal discovery algorithms to incorrectly remove causal dependencies. In this paper, we argue that hard interventions contain information about the presence/absence of causal linkage that is overlooked in the first stage of structure discovery. We show that a mild assumption -- called intervention-immediacy faithfulness -- that allows cancellations, is sufficient to nonparametrically identify causal structures with hard interventions. These results position interventions as the primary carriers of information about causal structure, which should take precedence over conditional independence testing. To flip the paradigm, we also specify equivalence classes when the identification criteria are not met due to limitations in the scope of interventions.

Sparse causal discovery calls for methods that exploit graph structure without estimating high-dimensional densities. We introduce Fisher Information Completion Search (FiCS), a source-first algorithm for additive noise models that uses one local Fisher score for both ordering and parent selection. Under regularity and nonconstant-parent conditions, we prove that a node's local Fisher information equals the noise Fisher information exactly when the conditioning set contains all parents, provided that it contains no descendants. This Fisher parent completion identifies the parent set as the unique minimal Fisher completion. With a maximum conditioning set size $q$ at least the maximum indegree $d$, population FiCS queries marginals of at most $q+1$ variables and recovers the true directed acyclic graph under a positive ordering margin. Bounded conditioning also has a population advantage: reducing $q$ toward $d$ cannot decrease, and can strictly increase, the ordering margin. A growing non-Gaussian family separates local Fisher selection from conditional-variance and leaf-first Fisher ordering. For the regularized kernel Stein estimator, we establish high-dimensional DAG consistency under $q\{1+\log(p/q)\}+\log p=o(n)$, uniform Fisher separation, local approximation, and compatible ridge and parent penalty parameters. Experiments show the strongest gains when $n$ is small relative to $p$, quantify the effect of the conditioning size, and demonstrate competitive reference-graph recovery on three real-data benchmarks.