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Equation 17 · Probing, Sparse Autoencoders, Patching, and Steering: The Main Interpretability Methods, Compared

What does this equation mean?

score(C)=mpatch(C)−mcorruptmclean−mcorrupt,\text{score}(\mathcal C) = \frac{m_{\text{patch}}(\mathcal C) - m_{\text{corrupt}}}{m_{\text{clean}} - m_{\text{corrupt}}},

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withm_patch(mathcal C) - m_corrupt
Divide bym_clean - m_corrupt
This relates toscore(mathcal C)
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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CC

Symbol C

copied from one run into the other.

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mpatchm_{\text{patch}}

Symbol m_patch

that same metric after the activations of component set C\mathcal C are copied from one run into the other.

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mcorruptm_{\text{corrupt}}

Symbol m_corrupt

a chosen behavioural metric measured on the clean and corrupted runs.

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mcleanm_{\text{clean}}

Symbol m_clean

a chosen behavioural metric measured on the clean and corrupted runs.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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subtraction

subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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mpatch(C)−mcorruptm_{\text{patch}}(\mathcal C) - m_{\text{corrupt}}

Numerator: m_patch(mathcal C) - m_corrupt

The complete quantity above the fraction bar.

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mclean−mcorruptm_{\text{clean}} - m_{\text{corrupt}}

Denominator: m_clean - m_corrupt

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

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

The general operation, activation patching, is usually reported as a normalised score rather than a raw difference, isolating exactly how much of the gap between clean and corrupted behaviour a given component set closes: score(C)=mpatch(C)−mcorruptmclean−mcorrupt\text{score}(\mathcal C) = \frac{m_{\text{patch}}(\mathcal C) - m_{\text{corrupt}}}{m_{\text{clean}} - m_{\text{corrupt}}}. where mcleanm_{\text{clean}} and mcorruptm_{\text{corrupt}} are a chosen behavioural metric measured on the clean and corrupted runs, and mpatch(C)m_{\text{patch}}(\mathcal C) is that same metric after the activations of component set C\mathcal C are copied from one run into the other. A score near one means patching C\mathcal C alone restores nearly all of the clean behaviour; a score near zero means it restores almost none.

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Sources cited in the article section

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