What does this equation mean?

The underlying reason none of these approaches resolves the problem is structural. A published score for model i on benchmark b can be decomposed, at least conceptually, as $\hat{s}_{i,b} = g_b(\theta_i) + \delta_{i,b} + \varepsilon_{i,b}$. where $\theta_i$ is some unobservable underlying ability vector for model i , $g_b$ is the mapping from ability to expected score that is specific to benchmark b and not shared across benchmarks, $\delta_{i,b}$ is a contamination or leakage term specific to that model-benchmark pair, and $\varepsilon_{i,b}$ is sampling and decoding noise. Because $g_b$ differs across benchmarks by construction — a multiple-choice knowledge test and a pairwise human-preference vote are not measuring the same projection of $\theta_i$ — there is no aggregation operator that turns a vector of $\hat{s}_{i,b}$ values into a single number without an additional, unverified assumption about how the $g_b$ functions relate to one another. HELM’s response is to refuse the aggregation and publish the matrix. Epoch’s response is to aggregate anyway, for a stated and narrower purpose. Neither is wrong; neither solves the general problem, because the general problem — finding the true, benchmark-independent $\theta_i$ from observed $\hat{s}_{i,b}$ — remains open.