Belief-Entropy Wasserstein Loss

Integrate this as a replacement for ordinary cross-entropy in a classifier or safety-value network whose outcomes have a meaningful geometry, such as discretized severity, distance-to-failure, or tool-risk levels. Use $K$ stochastic prediction heads producing $p_k(y\mid x)$ and a gate posterior $b_k(x)$. First form $\bar p(y\mid x)=\sum_kb_k(x)p_k(y\mid x)$ and $\epsilon=\beta[-\sum_kb_k\log(b_k+10^{-8})]$. For an $M$-class target, define per-outcome loss $f_i=-\log(\bar p_i+10^{-8})$ and a metric matrix $D_{ij}=d(i,j)$. Compute L = max_ij(abs(f[i]-f[j])/(D[i,j]+1e-8)); for each $\lambda$ on a grid from zero to $L$, compute envelope[i] = min_j(f[j] + lambda*D[j,i]) and dual[lambda] = sum_i(bar_p[i]*envelope[i]) - lambda*epsilon; use the largest dual value as the robust loss. Backpropagate through the selected grid point, or use a soft maximum over dual values for smoother gradients. The mathematical ingredients are the Wasserstein dual and entropy radius; the ensemble, posterior calibration, metric, and $\beta$ are empirical. Start with CIFAR-10-C or a three-level corruption-severity classifier, comparing cross-entropy, label smoothing, fixed-radius Wasserstein DRO, and entropy-calibrated DRO. Track clean accuracy, worst-corruption accuracy, expected calibration error, and robust loss versus ensemble entropy. The desired result is improved high-uncertainty calibration and corruption performance with little clean-accuracy loss, while the robust correction becomes smaller as ensemble agreement increases.