What does this equation mean?

The standard instrument is activation patching. Run the model once on a clean input and once on a deliberately corrupted variant of it — a single token changed, a name swapped — caching every intermediate activation from both runs. Then run the model a third time on the corrupted input, but with one chosen component’s activation overwritten by the value it took during the clean run, and measure the resulting change in some scalar behavioural metric m , typically the gap between the correct answer’s logit and a specific incorrect competitor’s logit: $\mathrm{Patch}(c) = m\Big(M\big(x_{\mathrm{corrupt}};\ a_c \leftarrow a_c^{\mathrm{clean}}\big)\Big) - m\big(M(x_{\mathrm{corrupt}})\big)$. This particular direction — a clean value spliced into an otherwise corrupted run — is called denoising: it measures how much of the clean behaviour component c can restore on its own. The complementary direction, a corrupted value spliced into an otherwise clean run, is noising, and it measures how much clean behaviour depends on c specifically. The two ask different questions, and a careful study reports both, together with the exact corruption and the exact metric used, because Zhang and Nanda showed that changing either of those methodological choices can change which components a patching study identifies as important for the same behaviour in the same model — a localisation claim, they establish, is a claim relative to a stated protocol, not a free-standing fact about the network [ 11 ] .