A key quantity of interest is the mutual information and generalizations thereof, including conditional mutual information, multivariate mutual information, total correlation and directed information.

It is not clear, however, whether these theoretical results apply to actual distributions such as images.

Generative models trained using self-supervision of tokenized electronic health record (EHR) timelines show promise for clinical outcome prediction. This is typically done using Monte Carlo simulation for future patient trajectories. However, existing approaches suffer from three key limitations: sparse estimate distributions that poorly differentiate patient risk levels, extreme computational costs, and high sampling variance. We propose two new estimators: the Sum of Conditional Outcome Probability Estimator (SCOPE) and Risk Estimation from Anticipated Conditional Hazards (REACH), that leverage next-token probability distributions discarded by standard Monte Carlo. We prove both estimators are unbiased and that REACH guarantees variance reduction over Monte Carlo sampling for any model and outcome. Empirically, on hospital mortality prediction in MIMIC-IV using the ETHOS-ARES framework, SCOPE and REACH match 100-sample Monte Carlo performance using only 10-11 samples (95% CI: [9,11]), representing a ~10x reduction in inference cost without degrading calibration. For ICU admission prediction, efficiency gains are more modest (~1.2x), which we attribute to the outcome's lower "spontaneity," a property we characterize theoretically and empirically. These methods substantially improve the feasibility of deploying generative EHR models in resource-constrained clinical settings.

Large language models (LLMs) exhibit complementary strengths in various tasks, motivating the research of LLM ensembling.However, existing work focuses on training an extra reward model or fusion model to select or combine all candidate answers, posing a great challenge to the generalization on unseen data distributions.Besides, prior methods use textual responses as communication media, ignoring the valuable information in the internal representations.In this work, we propose a training-free ensemble framework \textsc{DeePEn}, fusing the informative probability distributions yielded by different LLMs at each decoding step.Unfortunately, the vocabulary discrepancy between heterogeneous LLMs directly makes averaging the distributions unfeasible due to the token misalignment.To address this challenge, \textsc{DeePEn} maps the probability distribution of each model from its own probability space to a universal \textit{relative space} based on the relative representation theory, and performs aggregation.Next, we devise a search-based inverse transformation to transform the aggregated result back to the probability space of one of the ensembling LLMs (main model), in order to determine the next token.We conduct extensive experiments on ensembles of different number of LLMs, ensembles of LLMs with different architectures, and ensembles between the LLM and the specialist model.Experimental results show that (i) \textsc{DeePEn} achieves consistent improvements across six benchmarks covering subject examination, reasoning, and knowledge, (ii) a well-performing specialist model can benefit from a less effective LLM through distribution fusion, and (iii) \textsc{DeePEn} has complementary strengths with other ensemble methods such as voting.

We propose a novel Frank-Wolfe (FW) procedure for the optimization of infinite-dimensional functionals of probability measures - a task which arises naturally in a wide range of areas including statistical learning (e.g.

End-to-End Neural Diarization (EEND) systems produce frame-level probabilistic speaker activity estimates, yet since evaluation focuses primarily on Diarization Error Rate (DER), the reliability and calibration of these confidence scores have been largely neglected. When fusing multiple diarization systems, DOVER-Lap remains the only established approach, operating at the segment level with hard decisions. We propose working with continuous probability outputs, which enables more sophisticated fusion and calibration techniques that can leverage model uncertainty and complementary strengths across different architectures. This paper presents the first comprehensive framework for calibrating and fusing EEND models at the probability level. We investigate two output formulations (multilabel and powerset representations) and their impact on calibration and fusion effectiveness. Through extensive experiments on the CallHome two-speaker benchmark, we demonstrate that proper calibration provides substantial improvements even for individual models (up to 19% relative DER reduction), in some cases mitigating the absence of domain adaptation. We reveal that joint calibration in powerset space consistently outperforms independent per-speaker calibration, that fusion substantially improves over individual models, and that the Fuse-then-Calibrate ordering generally outperforms both calibrating before fusion and uncalibrated fusion while requiring calibration of only a single combined model. Our best configuration outperforms DOVER-Lap in terms of DER while providing reliable confidence estimates essential for downstream applications. This work proposes best practices for probability-level fusion of EEND systems and demonstrates the advantages of leveraging soft outputs over hard decisions.

In this paper, we address the graph matching problem.

A key quantity of interest is the mutual information and generalizations thereof, including conditional mutual information, multivariate mutual information, total correlation and directed information.

These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing kernel Hilbert spaces (RKHS), and Hilbert-Schmidt operators, emphasizing their role in statistical estimation and representation of probability measures. Classical concepts such as covariance, regression, and information measures are revisited through the lens of Hilbert space geometry. We also introduce kernel density estimation, kernel embeddings of distributions, and the Maximum Mean Discrepancy (MMD). The exposition is designed to serve as a foundation for more advanced topics, including Gaussian processes, kernel Bayesian inference, and functional analytic approaches to modern machine learning.