financial risk modeling teams

Abstract: Bayesian state estimation for high-dimensional nonlinear dynamical systems entails a fundamental tension between statistical fidelity and computational tractability, as particle weights can collapse, while Gaussian ensemble updates can miss non-Gaussian posterior structure. Score-based diffusion filters offer a sampling-based alternative, but existing training-free score filters often rely on heuristic likelihood corrections, which can compromise posterior accuracy by neglecting uncertainty about the system state associated with each noisy reverse particle. To address these issues, we propose AECSF, a training-free adaptive ensemble conditional score filter. AECSF constructs an analytically tractable score estimator from the conditional Tweedie identity, which recasts noisy posterior score estimation as estimating the conditional mean of the system state given a noisy reverse particle and the observation. To estimate these conditional means efficiently, AECSF employs a shared adaptive weighted proposal ensemble, while particle-specific conditional weights yield an estimate for each noisy reverse particle without separate proposal sampling. The proposal ensemble is updated using reverse-particle information within the same reverse-diffusion run to improve conditional-mean estimation. Theoretically, we characterize when a fixed weighted proposal measure yields the exact noisy posterior score. Under stated assumptions, we establish a bound relating conditional-mean estimation errors to reverse-sampling endpoint error. Numerical experiments demonstrate that AECSF improves the accuracy of posterior sampling and nonlinear filtering in high-dimensional problems with limited forecast ensembles.

Abstract: Numerical embeddings for click-through rate (CTR) prediction are built on a convenient but restrictive premise: a scalar has one representation. This premise conflates where a value lies with what it means for the current sample. On the Criteo validation split, the same numerical interval carries residual click evidence with opposite signs across categorical and numerical contexts, even after additive main effects are removed. Production pipelines compound this mismatch because externally normalized features require transformations and statistics to remain synchronized between training and serving. We introduce ScalarLens, a numerical embedding that preserves what a value is while adapting how it should be interpreted. A monotone local mesh constructs a stable coordinate from the focal scalar alone; bounded low-rank dynamics then produce a contextual response without moving that coordinate or replacing categorical tokens and the CTR backbone. In a 1,539-run primary evaluation covering 19 representations, three datasets, nine backbones, and three seeds, ScalarLens ranks first in 25 of 27 settings on original numerical scales and second in the remaining two. Matched ablations show that scale correction, additional local capacity, and generic conditioning do not reproduce the gain. A controlled study further recovers categorical, numerical, and mixed response mechanisms under context shift while the focal coordinate remains exactly invariant. A complete rerun under shared standardization retains significant advantages over DEER, DAES, and NaryDis, showing that the result is not explained by tolerance to raw scales alone. ScalarLens therefore recasts numerical embedding as a measurement problem: coordinates belong to values, while predictive responses belong to values in context.