Distribution Estimation with Side Information

2026-01-13

AI Summary

This study addresses the problem of improving the accuracy of discrete distribution estimation by leveraging semantic and other auxiliary information. It presents the first systematic modeling and theoretical quantification of how such auxiliary information reduces squared error risk under two settings: a local model, where the unknown distribution is close to a known prior, and a partial-order model, where the alphabet is partitioned into high- and low-probability subsets. The proposed framework integrates statistical learning theory, risk analysis, and semantic representations—such as word embeddings—to construct distributional neighborhoods and partial-order structures. Experiments on both natural language and synthetic data demonstrate that the approach significantly lowers estimation error, with empirical results closely aligning with theoretical predictions.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Natural Language Processing: Sentence-level Semantics, Textual Inference, etc.

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We consider the classical problem of discrete distribution estimation using i.i.d. samples in a novel scenario where additional side information is available on the distribution. In large alphabet datasets such as text corpora, such side information arises naturally through word semantics/similarities that can be inferred by closeness of vector word embeddings, for instance. We consider two specific models for side information--a local model where the unknown distribution is in the neighborhood of a known distribution, and a partial ordering model where the alphabet is partitioned into known higher and lower probability sets. In both models, we theoretically characterize the improvement in a suitable squared-error risk because of the available side information. Simulations over natural language and synthetic data illustrate these gains.

Problem

• distribution estimation
• side information
• discrete distribution
• large alphabet
• word embeddings

Innovation

• distribution estimation
• side information
• word embeddings
• squared-error risk
• large alphabet