Paul Schneider

Michael Multerer, Paul Schneider, Rohan Sen

We seek to extract a small number of representative scenarios from large panel data that are consistent with sample moments. Among two novel algorithms, the first identifies scenarios that have not been observed before, and comes with a scenario-based representation of covariance matrices. The second proposal selects important data points from states of the world that have already realized, and are consistent with higher-order sample moment information. Both algorithms are efficient to compute and lend themselves to consistent scenario-based modeling and multi-dimensional numerical integration that can be used for interpretable decision-making under uncertainty. Extensive numerical benchmarking studies and an application in portfolio optimization favor the proposed algorithms.

Michael Multerer, Paul Schneider, Chiara Segala

We propose a scalable and theoretically grounded low-rank conditional expectation model for recursive Monte Carlo optimal stopping problems, in particular American option pricing. Our method reformulates the estimation of continuation values as a learning problem in a reproducing kernel Hilbert space, in which the conditional expectation is represented as a linear operator acting on future payoffs. This perspective yields an offline-online decomposition: the operator is learned once from simulated data and subsequently reused across all exercise dates, eliminating the need to recompute regression models at each step of the backward recursion. We establish convergence guarantees and derive bounds quantifying the approximation errors across exercise dates. Numerical experiments demonstrate the speed and accuracy of the proposed approach relative to extant methods.

Paul Schneider, Amalie Schramm

Structured deliberation has been found to improve the performance of human forecasters. This study investigates whether a similar intervention, i.e. allowing LLMs to review each other's forecasts before updating, can improve accuracy in large language models (GPT-5, Claude Sonnet 4.5, Gemini Pro 2.5). Using 202 resolved binary questions from the Metaculus Q2 2025 AI Forecasting Tournament, accuracy was assessed across four scenarios: (1) diverse models with distributed information, (2) diverse models with shared information, (3) homogeneous models with distributed information, and (4) homogeneous models with shared information. Results show that the intervention significantly improves accuracy in scenario (2), reducing Log Loss by 0.020 or about 4 percent in relative terms (p = 0.017). However, when homogeneous groups (three instances of the same model) engaged in the same process, no benefit was observed. Unexpectedly, providing LLMs with additional contextual information did not improve forecast accuracy, limiting our ability to study information pooling as a mechanism. Our findings suggest that deliberation may be a viable strategy for improving LLM forecasting.

Damir Filipovic, Paul Schneider

We introduce kernel density machines (KDM), a nonparametric estimator of a Radon--Nikodym derivative, based on reproducing kernel Hilbert spaces. KDM applies to general probability measures on countably generated measurable spaces under minimal assumptions. For computational efficiency, we incorporate a low-rank approximation with precisely controlled error that grants scalability to large-sample settings. We provide rigorous theoretical guarantees, including asymptotic consistency, a functional central limit theorem, and finite-sample error bounds, establishing a strong foundation for practical use. Empirical results based on simulated and real data demonstrate the efficacy and precision of KDM.

Damir Filipovic, Paul Schneider

We develop a nonparametric, kernel-based joint estimator for conditional mean and covariance matrices in large and unbalanced panels. The estimator is supported by rigorous consistency results and finite-sample guarantees, ensuring its reliability for empirical applications. We apply it to an extensive panel of monthly US stock excess returns from 1962 to 2021, using macroeconomic and firm-specific covariates as conditioning variables. The estimator effectively captures time-varying cross-sectional dependencies, demonstrating robust statistical and economic performance. We find that idiosyncratic risk explains, on average, more than 75% of the cross-sectional variance.

Damir Filipovic, Michael Multerer, Paul Schneider

We develop a new framework for estimating joint probability distributions using tensor product reproducing kernel Hilbert spaces (RKHS). Our framework accommodates a low-dimensional, normalized and positive model of a Radon--Nikodym derivative, which we estimate from sample sizes of up to several millions, alleviating the inherent limitations of RKHS modeling. Well-defined normalized and positive conditional distributions are natural by-products to our approach. Our proposal is fast to compute and accommodates learning problems ranging from prediction to classification. Our theoretical findings are supplemented by favorable numerical results.