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Equation 4 · Part 7 · Measuring Frontier Models: Contamination, Variance, and What a Score Can Support

fraction

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}} .
fraction

What this part means

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

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

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 fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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