Everything, Everywhere, All at Once: Is Mechanistic Interpretability Identifiable?

As AI systems are increasingly deployed in high-stakes applications, ensuring their interpretability is essential. Mechanistic Interpretability (MI) aims to reverse-engineer neural networks by extracting human-understandable algorithms embedded within their structures to explain their behavior. This work systematically examines a fundamental question: for a fixed behavior to explain, and under the criteria that MI sets for itself, are we guaranteed a unique explanation? Drawing an analogy with the concept of identifiability in statistics, which ensures the uniqueness of parameters inferred from data under specific modeling assumptions, we speak about the identifiability of explanations produced by MI.We identify two broad strategies to produce MI explanations: (i) "where-then-what", which first identifies a subset of the network (a circuit) that replicates the model's behavior before deriving its interpretation, and (ii) "what-then-where", which begins with candidate explanatory algorithms and searches in the activation subspaces of the neural model where the candidate algorithm may be implemented, relying on notions of causal alignment between the states of the candidate algorithm and the neural network. We systematically test the identifiability of both strategies using simple tasks (learning Boolean functions) and multi-layer perceptrons small enough to allow a complete enumeration of candidate explanations. Our experiments reveal overwhelming evidence of non-identifiability in all cases: multiple circuits can replicate model behavior, multiple interpretations can exist for a circuit, several algorithms can be causally aligned with the neural network, and a single algorithm can be causally aligned with different subspaces of the network.We discuss whether the unicity intuition is necessary. One could adopt a pragmatic stance, requiring explanations only to meet predictive and/or manipulability standards. However, if unicity is considered essential, e.g., to provide a sense of understanding, we also discuss less permissive criteria. Finally, we also refer to the inner interpretability framework that demands explanations to be validated by multiple complementary criteria. This work aims to contribute constructively to the ongoing effort to formalize what we expect from explanations in AI.