Accurate and well-calibrated Machine Learning (ML) models are mandatory in high-stakes settings, yet effective multiclass calibration remains challenging: global approaches assume calibration errors are homogeneous across the latent space, while local methods often rely on latent-space dimensionality reduction, which leads to information loss. To address these issues, we propose a compositional approach to multiclass calibration, where region-specific calibration maps are constructed from shared codeword-dependent factors. We instantiate this idea via Vector Quantization (VQ), which induces a structured partition of the representation space, and an indexed parameterization of Dirichlet concentrations that enables parameter sharing across regions. Our approach learns heterogeneous calibration maps that generalize well even to sparse regions of the latent space. Experiments on benchmark datasets show significant improvements in local calibration while maintaining competitive global calibration and predictive performance.

Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In applications, however, the measures of interest are almost never known precisely, calling for a theory of optimal transport that accounts for statistical uncertainty. We construct such a framework, lifting the classical theory to the setting of random probability measures. We introduce the $L^2$ over Wasserstein space establishing that it inherits the formal Riemannian structure of the Wasserstein space by characterising distances and geodesic geometry. The structure induces random flows with Wasserstein gradient flow sample paths, making it the natural extension of the Wasserstein space which allows for random gradient flow dynamics. We ensemble statistical convergence results of the optimal transport machinery using the empirical measure within the $L^2$ over Wasserstein framework. Moreover, in the setting of Bayesian non-parametrics, we refine Schwartz's consistency theorem to the Wasserstein topology and deduce posterior convergence of the same machinery in the $L^2$ over Wasserstein space. We demonstrate that the growing theory of random token sampling for transformer models using self-attention flow paths can be embedded into the our framework. The results provide a unified treatment of random optimal transport and its consequences for principled inference and generative modelling under the statistical uncertainty of random sampling.

Bayesian latent space models offer a principled approach to network representation, but rely on correct specification of both geometry and link function. Real-world networks often violate these assumptions, exhibiting geometric mismatch and structural anomalies that break standard metric properties. We show that such misspecification pushes the data-generating distribution outside the model class, causing Bayesian inference to become overconfident and poorly calibrated. To address this, we propose a generalized posterior framework for random geometric graphs. We introduce Link-Sequential R-SafeBayes, a method that exploits dyadic conditional independence to estimate prequential risk and adaptively tune posterior regularization. Experiments on synthetic and real-world networks demonstrate improved calibration, better link prediction performance, and a reliable criterion for selecting latent geometries across Euclidean, spherical, and hyperbolic spaces.

We introduce DeRegiME -- Deep Regime Mixture of Experts -- a direct multi-horizon probabilistic forecaster that separates latent uncertainty regimes from the underlying signal and softly assigns each forecast location to learned recurring regimes using a sparse variational Gaussian process (GP) whose nonstationary regime-mixing kernel and Student-t likelihood combine per-regime sub-kernels and noise processes via a shared gate. This yields a single sparse-GP posterior, not a mixture of GP experts. DeRegiME addresses a key limitation of neural forecasters: point forecasts discard residual uncertainty, and probabilistic heads -- whether single marginals, uninterpreted mixtures, quantile sets, or diffusion samples -- rarely expose the regime structure of the residual. Yet distribution shift in noisy heteroskedastic time series may be abrupt, gradual, or horizon-dependent and often appears in residual uncertainty rather than the conditional mean. DeRegiME yields an interpretable mean-residual-noise decomposition with a direct-sum feature-space representation that anchors regimes as clusters of residual similarity whose transitions surface as implicit changepoints. The effective number of regimes is pruned by the stick-breaking gate. We prove kernel validity and predictive-density propriety, and across ten benchmarks and three encoder grids DeRegiME improves negative log predictive density (NLPD) by 20.3% over the strongest encoder-matched baseline, a DeepAR/GluonTS-style dynamic Student-t head, with parallel gains on CRPS (3.0%) and MSE (4.7%). Improvements are consistent across all datasets, which span abrupt, gradual, and seasonal shifts.

Sequential prediction is challenging in regimes of delayed disambiguation, where early observations are ambiguous and multiple latent explanations remain plausible until sufficient evidence accumulates. Standard approaches based on marginal inference struggle in this setting, either collapsing uncertainty prematurely or failing to recover once informative evidence arrives. We introduce EviTrack, a test-time inference framework that operates over latent trajectories rather than marginal states. EviTrack maintains a set of competing trajectory hypotheses and applies evidence- and likelihood-ratio-based selection to delay commitment until supported by data, drawing inspiration from hypothesis management in multiple hypothesis tracking and track-before-detect. To evaluate this setting, we construct a controlled synthetic benchmark with known latent ground truth that explicitly exhibits delayed disambiguation. At matched inference budget, EviTrack substantially outperforms sampling-based baselines, achieving faster post-disambiguation recovery. These results show that, in delayed disambiguation regimes, moderate trajectory-level selection is more effective than increasing sampling coverage, highlighting selection over sampling as a key principle for reliable sequential inference.

We investigate the asymptotic properties of the Lévy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the Lévy Adaptive Regression Kernel (LARK) model. LABS applies splines of varying degrees with independently defined knots, yielding a flexible model class capable of adapting to irregular and locally structured features of the true function. Within the nonparametric regression framework with univariate random design and Gaussian errors, we establish that the LABS posterior contracts around the true function in Besov classes at nearly minimax-optimal rates, up to a logarithmic factor, while adapting automatically to unknown smoothness. This study contributes to filling a gap in the literature, where theoretical results on posterior contraction of the LARK model in Besov spaces remain scarce. Simulation experiments on standard test functions in Besov spaces, including Blocks, Bumps, HeaviSine, and Doppler, complement the theoretical results and demonstrate the practical utility of LABS.

Bayesian optimization (BO) selects evaluation points for expensive black-box objectives using Gaussian process (GP) predictive distributions. Kernel choice and hyperparameter selection can lead to miscalibrated predictive distributions and an inappropriate exploration-exploitation trade-off. For minimization, sampling criteria such as expected improvement (EI) depend on the predictive distribution below the current best value, so lower-tail miscalibration directly affects the sampling decision. This article studies goal-oriented calibration of GP predictive distributions below a low threshold $t$ in the noiseless setting, for standard GP models with hyperparameters selected by maximum likelihood. A framework for predictive reliability below $t$ is introduced, based on two notions of spatial calibration: occurrence calibration over the design space and thresholded $μ$-calibration on sublevel sets of the form $\{x\in\mathbb{X}, f(x)\le t\}$. Building on this framework, we propose tcGP, a post-hoc method that calibrates GP predictive distributions below~$t$, and we show that the resulting EI-based global optimization algorithm remains dense in the design space. Experiments on standard benchmarks show improved lower-tail calibration and BO performance relative to standard GP models and globally calibrated GP models.

Deep neural networks achieve high accuracy on image classification tasks. Yet, they often produce overconfident predictions as which fail to express epistemic uncertainty, and frequently violate logical or structural constraints present in the data. These limitations are particularly pronounced in hierarchical classification, where predictions across fine and coarse levels must remain coherent. We propose, for the first time, a unified neurosymbolic and epistemic modelling framework that augments Swin Transformers with focal set reasoning and differentiable fuzzy logic. Rather than treating labels as isolated categories, our method induces data-driven focal sets within the learnt embedding space, which helps capture epistemic uncertainty over multiple plausible fine-grained classes. These focal sets form the basis of a belief-theoretic layer that uses fuzzy membership functions and t-norm conjunctions to encourage consistency between fine- and coarse-grained predictions. A learnable loss further balances calibration, mass regularisation, and logical consistency, allowing the model to adaptively trade off symbolic structure with data-driven evidence. In experiments on hierarchical image classification, our framework maintains accuracy on par with transformer baselines while providing more calibrated and interpretable predictions, reducing overconfidence and enforcing high logical consistency across hierarchical outputs. Our experimental results show that combining focal set reasoning with fuzzy logic provides a practical step toward deep learning models that are both accurate and epistemically aware.

Diffusion and flow-based models are ubiquitously used for generative modelling and density estimation. They admit a deterministic probability flow ordinary differential equation (PF-ODE), analogous to continuous normalizing flows (CNFs), which describes the transport of the probability mass. Obtaining the likelihood from these models is of interest to many workflows, especially Bayesian analysis, and requires solving the trace of the Jacobian to compute the divergence of the learned PF-ODE, which is either $\mathcal{O}(D^2)$ to compute exactly or $\mathcal{O}(D)$ with a noisy estimate. We introduce StAD, a new distillation method to predict and learn the divergence of the PF-ODE using the Langevin-Stein operator without ever computing the Jacobian. We show that our method is competitive with the Hutchinson and Hutch++ on CIFAR-10, ImageNet and other density estimation tasks, consistently improving the variance and speed of the likelihood predictions compared to the Hutchinson. We additionally show our method will generalize to a varied class of generative models, and show that under some regularity conditions these learned vector fields can be made to satisfy the Stein class.

A central obstacle in nonlinear Bayesian filtering is representing the belief distribution. Moment-based filters address this by propagating polynomial moments and reconstructing a density from them. Recent work completes the predict-update loop via the maximum-entropy (MaxEnt) principle, but each step requires the partition function and its gradient, both $n$-dimensional integrals whose cost scales exponentially, restricting the demonstrated MaxEnt moment filtering to $n \le 4$. We avoid the partition function entirely by combining score matching with Stein's identity. In our setting, score matching reduces the density fit to a single linear solve whose coefficients are assembled directly from the propagated moments. The same parameters then drive Stein's identity to close the moment hierarchy during prediction and to recover posterior moments after each Bayesian update, keeping the full predict-update loop free of partition function evaluation. The resulting Score Kalman Filter (SKF) reduces to the classical information-form Kalman filter as a special case and performs every step through linear algebra. On nonlinear coupled-oscillator networks, the SKF runs through $n=20$ and reports lower RMSE than the EKF, UKF, EnKF, and particle-filter baselines on the tested synthetic benchmarks.