How should a verifier's vector report be scored?

What it found

For a report of an expected drift r ∈ ℝ^d with noise covariance Σ, every quadratic Bregman score −(y−r)^TA(y−r) is proper and charges a misreport D exactly D^TAD. Against a trusted reference the per-task gap has exact mean D^TAD and variance 4D^TAΣAD for any noise law, and Cauchy–Schwarz shows A=Σ⁻¹ has the best signal to noise for every deviation at once, needing 4z²/(D^TΣ⁻¹D) tasks. An isotropic score has efficiency at least 4κ/(1+κ)² (Kantorovich), attained by a deviation split between the extreme eigendirections: 25× more tasks at condition number 100, and power 0.29 against 0.80 at 48 tasks in simulation. Per-layer normalisation alone barely helps; with Σ estimated from m reference draws, efficiency is 0.61 at m=2d and 0.95 at m=100. Quadratic scores, trusted reference, finite covariance.

A figure from its own output

Paper, code and results

• README and method
• White paper
• Experiments and results
• Code
• Tests

More in Proper scoring & elicitation

• What does a verifier's score gap cost in decisions?From a verifier's excess proper score to a certified cap on accept/audit/slash decision regret: exact calibration functions (Brier ε², log KL), linear transfer only for non-proper hinge, audit action tightens the guarantee.
• Which surrogate should train a verifier that can escalate?Polyhedral surrogates for accept/reject/escalate verifiers: minimiser embeds the Bayes action, linear regret transfer with constant exactly 2d, logistic transfer ratio 2d(1-d)/δ, and a negative SGD result.
• How do you pay a verifier whose report decides the job?Scoring verifiers whose report decides acceptance and whose outcome is seen only on acceptance: a truthful reserve exists iff the score centre τ_s ≥ τ_d (unique c0 = 1+τ_d²−2τ_dτ_s), plain Brier fails above τ_d = ½, recentring fixes it at rent τ³/3, exploration costs 1/ε payment variance.
• What if the referee itself is noisy?Scoring verifiers against a noisy judge: raw proper scores elicit e0+γp and shrink Brier incentives by γ², can reverse rankings (η*=0.25), the unbiased surrogate score is exactly proper at a 1/γ payment spread and 1/γ² sample cost, and misestimated rates give a closed-form bias.
• Can audits be predictable without corrupting scores?Scoring when truth exists only on audited tasks: report-dependent audit rates make naive scoring improper (shift −bp(1−p)/(2(a+bp)), full hiding for steep rates), inverse-propensity weighting is exactly proper for any audit rule, and Neyman allocation gains only 3–11% variance.
• Can a calibration test certify a verifier?Calibration tests are gameable: an information-free Foster–Vohra/Blackwell hedger keeps expected ECE ≤ √(K+1)(1/(2K)+1/√T) against any outcome sequence (measured 0.016, Brier 0.254 vs 0 informed), so slashing must use a resolution-sensitive score; a Brier-gap test detects it in 100/100 runs.

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Other research topics

Proper scoring & elicitation28Effort & peer prediction9Verification games & slashing19Detection & tolerances14Markets as routers13Prediction-market design9Paying contributors8Decentralised training systems23Digital twins & sim-to-real164LLM chip design loops40Robot manufacturing loops53Ambient AI assistant46Self-replicating manufacturing34Simulation-first design64Engineered clothing45Task-specific AI accelerators40Algorithmic chip-design loops42Open-source purpose-built robots44The software-engineering loop43Adversarial human imitation41Open-source residential construction41Autonomous multi-angle capture34Sensing, imaging and 3D31Energy supply and dense energy storage39Robot societies among people31Agentic co-design of low-cost robots27Laundry-hanging robots25Machine learning on qualitative data23Robots on the building site27Civilizations & group psychology1History & human sciences1

Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.