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Equation 25 · Part 19 · Probing, Sparse Autoencoders, Patching, and Steering: The Main Interpretability Methods, Compared

Starting index or lower bound: i in N

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,
i∈Ni \in N

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

Its job in the formula

i in N appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the article section

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