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Equation 4 · Part 3 · Measuring Frontier Models: Contamination, Variance, and What a Score Can Support

Symbol i

s^=1n∑i=1n1 ⁣[ success on xi ],xi∼Dbench.\hat{s} = \frac{1}{n}\sum_{i=1}^{n} \mathbf{1}\!\left[\,\text{success on } x_i\,\right], \qquad x_i \sim \mathcal{D}_{\mathrm{bench}} .
ii

What this part means

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

Its job in the formula

i 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

Reframe the object. A benchmark score s^\hat{s} is an estimator of a population quantity s — the model’s success rate over some distribution of tasks D\mathcal{D} that somebody hopes resembles the work you actually have. Every property that makes an estimator trustworthy applies: s^=1n∑i=1n1 ⁣[ success on xi ],xi∼Dbench\hat{s} = \frac{1}{n}\sum_{i=1}^{n} \mathbf{1}\!\left[\,\text{success on } x_i\,\right], \qquad x_i \sim \mathcal{D}_{\mathrm{bench}} . The number is only as good as three things: whether Dbench\mathcal{D}_{\mathrm{bench}} resembles D\mathcal{D} , whether the xix_i are genuinely held out, and whether the indicator is measured with enough repetition to characterise its spread. All three fail routinely, and they fail in different directions.

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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