# Uncertainty Quantification

Online conformal risk control has assumed each round's loss arrives immediately and intact, yet real systems delay feedback and can track only a limited number of pending predictions, so some feedback is lost for good. CA-OCRC designs the tracking policy and the prediction sets together: each round draws a random proxy delay, is tracked while capacity allows until its feedback arrives or the proxy delay expires, and a two-level inverse probability weighting compensates for lost feedback. Risk bounds for arbitrary and stochastic delays follow, and optimizing the proxy delay distribution shows that capacity growing only logarithmically in the maximum delay suffices to match the convergence rate of OCRC with unlimited feedback.

Online conformal risk control has assumed each round's loss arrives immediately and intact, yet real systems delay feedback and can track only a limited number of pending predictions, so some feedback is lost for good. CA-OCRC designs the tracking policy and the prediction sets together: each round draws a random proxy delay, is tracked while capacity allows until its feedback arrives or the proxy delay expires, and a two-level inverse probability weighting compensates for lost feedback. Risk bounds for arbitrary and stochastic delays follow, and optimizing the proxy delay distribution shows that capacity growing only logarithmically in the maximum delay suffices to match the convergence rate of OCRC with unlimited feedback.

Bifidelity modeling uses cheap low-fidelity simulations to supplement scarce high-fidelity experiments, but the usual product is a single deterministic fit that says little about how trustworthy a prediction is. This work extends a single-fidelity Langevin sampling hypernetwork to bifidelity data, using a shared-latent data model that predicts both the low-fidelity response and the high-fidelity correction while a learned drift network generates parameter samples from the posterior score. On a synthetic hyperelastic case, conditional staged training cut the SPD-drift RMSE from 0.075 to 0.024 and the convex-potential drift from 0.081 to 0.017; on experimental and DNS data, predicting the high-fidelity response directly gave lower error than an additive discrepancy.

Bifidelity modeling uses cheap low-fidelity simulations to supplement scarce high-fidelity experiments, but the usual product is a single deterministic fit that says little about how trustworthy a prediction is. This work extends a single-fidelity Langevin sampling hypernetwork to bifidelity data, using a shared-latent data model that predicts both the low-fidelity response and the high-fidelity correction while a learned drift network generates parameter samples from the posterior score. On a synthetic hyperelastic case, conditional staged training cut the SPD-drift RMSE from 0.075 to 0.024 and the convex-potential drift from 0.081 to 0.017; on experimental and DNS data, predicting the high-fidelity response directly gave lower error than an additive discrepancy.

Standard neural processes treat inputs as noise-free observations, so sensor measurement error degrades both predictions and uncertainty estimates. DnNP makes the clean input a latent variable inside the neural-process framework and trains a denoiser jointly with meta-learning through variational inference, predicting directly from noisy inputs. On PM2.5 spatial interpolation from 34 air-quality sensors in Brookline, Massachusetts, DnNP reduces RMSE by up to 30% and NLL by up to 73% relative to standard neural processes.

Standard neural processes treat inputs as noise-free observations, so sensor measurement error degrades both predictions and uncertainty estimates. DnNP makes the clean input a latent variable inside the neural-process framework and trains a denoiser jointly with meta-learning through variational inference, predicting directly from noisy inputs. On PM2.5 spatial interpolation from 34 air-quality sensors in Brookline, Massachusetts, DnNP reduces RMSE by up to 30% and NLL by up to 73% relative to standard neural processes.

Array processing has long treated inter-sensor coherence loss as a nuisance, and conventional beamforming assumes a far-field point source with incoherent noise between sensors. Topophone2 models the array covariance as a mixture of energy distributions over slowness, describing signal and environmental noise with skew-normal and angular-harmonic distributions that have analytical characteristic functions, then computes the posterior by variational Bayesian inference. On a synthetic abyssal T-phase, adding a fourth hydrophone raises the information gain on the angular-harmonic noise coefficients by 142%, while the gain on signal parameters and SNR rises by less than 8%.

Array processing has long treated inter-sensor coherence loss as a nuisance, and conventional beamforming assumes a far-field point source with incoherent noise between sensors. Topophone2 models the array covariance as a mixture of energy distributions over slowness, describing signal and environmental noise with skew-normal and angular-harmonic distributions that have analytical characteristic functions, then computes the posterior by variational Bayesian inference. On a synthetic abyssal T-phase, adding a fourth hydrophone raises the information gain on the angular-harmonic noise coefficients by 142%, while the gain on signal parameters and SNR rises by less than 8%.

Evidential deep learning represents classification uncertainty through Dirichlet parameters, but the standard objective shows evidence contraction at high confidence. This work analyzes the optimization landscape of the regularized Dirichlet likelihood on a box domain, proves that the second-order sufficient conditions reduce to a scalar self-consistency derivative being positive, and gives an enumeration algorithm with a finite-convergence guarantee. On logits from a ResNet-18 trained on CIFAR-10, total evidence rises monotonically with confidence yet never exceeds a theoretical threshold near 19.75, while the active set shrinks from about 2.05 classes to one.

Evidential deep learning represents classification uncertainty through Dirichlet parameters, but the standard objective shows evidence contraction at high confidence. This work analyzes the optimization landscape of the regularized Dirichlet likelihood on a box domain, proves that the second-order sufficient conditions reduce to a scalar self-consistency derivative being positive, and gives an enumeration algorithm with a finite-convergence guarantee. On logits from a ResNet-18 trained on CIFAR-10, total evidence rises monotonically with confidence yet never exceeds a theoretical threshold near 19.75, while the active set shrinks from about 2.05 classes to one.

Across 160 classification and 81 regression datasets, tabular foundation model rankings change with the benchmark and the reliability metric, and regression models that already sit near nominal marginal coverage still under-cover in their worst feature slab, so model selection cannot rest on aggregate predictive accuracy alone.

Across 160 classification and 81 regression datasets, tabular foundation model rankings change with the benchmark and the reliability metric, and regression models that already sit near nominal marginal coverage still under-cover in their worst feature slab, so model selection cannot rest on aggregate predictive accuracy alone.