Bag of Coins: A Statistical Probe into Neural Confidence Structures

Modern neural networks often produce miscalibrated confidence scores and struggle to detect out-of-distribution (OOD) inputs, while most existing methods post-process outputs without testing internal consistency. We introduce the Bag-of-Coins (BoC) probe, a non-parametric diagnostic of logit coherence that compares softmax confidence $\hat p$ to an aggregate of pairwise Luce-style dominance probabilities $\bar q$, yielding a deterministic coherence score and a p-value-based structural score. Across ViT, ResNet, and RoBERTa with ID/OOD test sets, the coherence gap $\Delta=\bar q-\hat p$ reveals clear ID/OOD separation for ViT (ID ${\sim}0.1$-$0.2$, OOD ${\sim}0.5$-$0.6$) but substantial overlap for ResNet and RoBERTa (both ${\sim}0$), indicating architecture-dependent uncertainty geometry. As a practical method, BoC improves calibration only when the base model is poorly calibrated (ViT: ECE $0.024$ vs.\ $0.180$) and underperforms standard calibrators (ECE ${\sim}0.005$), while for OOD detection it fails across architectures (AUROC $0.020$-$0.253$) compared to standard scores ($0.75$-$0.99$). We position BoC as a research diagnostic for interrogating how architectures encode uncertainty in logit geometry rather than a production calibration or OOD detection method.