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Equation 25 · Part 13 · Probing, Sparse Autoencoders, Patching, and Steering: The Main Interpretability Methods, Compared

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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,
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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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