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Equation 25 · Part 15 · Probing, Sparse Autoencoders, Patching, and Steering: The Main Interpretability Methods, Compared

Denominator: |P|

vℓ=1∣P∣∑i∈Phℓ(xi+)  −  1∣N∣∑i∈Nhℓ(xi−),hℓ←hℓ+α vℓ,v_\ell = \frac{1}{|P|}\sum_{i \in P} h_\ell(x_i^{+}) \;-\; \frac{1}{|N|}\sum_{i \in N} h_\ell(x_i^{-}), \qquad h_\ell \leftarrow h_\ell + \alpha\, v_\ell,
∣P∣|P|

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

|P| occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

The construction is a mean-difference direction, added back into the residual stream with a tunable strength at inference time: vℓ=1∣P∣∑i∈Phℓ(xi+)  −  1∣N∣∑i∈Nhℓ(xi−),hℓ←hℓ+α vℓv_\ell = \frac{1}{|P|}\sum_{i \in P} h_\ell(x_i^{+}) \;-\; \frac{1}{|N|}\sum_{i \in N} h_\ell(x_i^{-}), \qquad h_\ell \leftarrow h_\ell + \alpha\, v_\ell. where P and N index matched positive and negative examples of the target behaviour, hℓh_\ell is the residual-stream activation at layer ℓ\ell , and α\alpha is a coefficient the operator sets by hand. Nothing in this construction requires understanding why the network represents the concept along this direction, only that adding it produces the intended behavioural shift.

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the article section

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