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Published equation contexts

p^θ(y∣aℓ)=σ ⁣(w⊤aℓ+b),θ={w,b}\hat p_\theta(y \mid a_\ell) = \sigma\!\left(w^{\top} a_\ell + b\right), \qquad \theta = \{w, b\}

Why this formula appears here

The first concrete operation in almost any interpretability project is the least glamorous: run the model forward on a batch of inputs, and at some chosen point in the computation — a residual-stream position, an attention head’s output, a particular MLP layer — copy the activation tensor out before it is overwritten by the next step of the forward pass. This is extraction, and it produces nothing on its own beyond a large table of vectors. What turns it into evidence is probing: fitting a small, separately trained classifier to predict some property of interest directly from those vectors, while the model’s own weights stay frozen. Formally, for an activation aℓa_\ell read out at layer ℓ\ell…

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p^θ\hat p_\theta

Symbol hat p_θ

hat p_θ is part of the quantity the equation computes from the expression on the right.

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aℓa_\ell

Symbol a_ell

aea_ell is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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p^θ(y∣aℓ)=σ ⁣(w⊤aℓ+b),θ={w,b},\hat p_\theta(y \mid a_\ell) = \sigma\!\left(w^{\top} a_\ell + b\right), \qquad \theta = \{w, b\},

Equation 4 · AI Research

How Mechanistic Interpretability Research Is Actually Done

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The first concrete operation in almost any interpretability project is the least glamorous: run the model forward on a batch of inputs, and at some chosen point in the computation — a residual-stream position, an attention head’s output, a particular MLP layer — copy the activation tensor out before it is overwritten by the next step of the forward pass. This is extraction, and it produces nothing on its own beyond a large table of vectors. What turns it into evidence is probing: fitting a small, separately trained classifier to predict some property of interest directly from those vectors, while the model’s own weights stay frozen. Formally, for an activation aℓa_\ell read out at layer ℓ\ell…

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