NP optimality of geometric neutral zones

Let $\mathbf{p}$ be a predicted class-probability vector, let $\mathbf{e}_{\hat{j}}$ be the one-hot vector for the predicted class $\hat{j}$, let $d_{\mathrm{FR}}$ denote the Fisher–Rao distance, and let $\lambda$ be the reliability-score parameter. Define the reliability score by

$$R=\exp\bigl(-\lambda\,d_{\mathrm{FR}}(\mathbf{p},\mathbf{e}_{\hat{j}})\bigr).$$

For the binary classification of an outcome as correct or error, the likelihood ratio is the ratio of the corresponding conditional probabilities.

NP optimality of geometric neutral zones. The reliability score induces a monotone likelihood-ratio ordering: whenever $R_1>R_2$,

$$\frac{\mathbb{P}(\text{correct}\mid R_1)}{\mathbb{P}(\text{error}\mid R_1)}>\frac{\mathbb{P}(\text{correct}\mid R_2)}{\mathbb{P}(\text{error}\mid R_2)}.$$

Consequently, threshold tests on $R$ are Neyman–Pearson optimal for the binary hypothesis test with $H_0$: error and $H_1$: correct.

The conjecture is presented as having strong empirical support, but its proof remains in progress. If true, it would provide a theoretical justification for using geometric reliability thresholds to construct optimal neutral zones in uncertainty-aware multiclass classification.