Simultaneous estimation of multiple discrete unimodal distributions under stochastic order constraints

2026-03-12

AI Summary

This work addresses the joint estimation of multiple discrete unimodal distributions when prior knowledge about their stochastic ordering is available. We propose the first mixed-integer convex quadratic optimization framework that explicitly incorporates stochastic order constraints derived from such prior knowledge into the joint estimation objective. By embedding these constraints, our method significantly improves estimation accuracy in small-sample regimes. Experimental results on both synthetic and real-world datasets demonstrate that, under limited data conditions, the proposed approach reduces the average Jensen–Shannon divergence by 2.2% (up to 6.3%) compared to existing methods, while achieving comparable performance when sufficient data are available. These findings confirm the effectiveness and robustness of the proposed framework across varying sample sizes.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of estimating multiple discrete unimodal distributions, motivated by search behavior analysis on a real-world platform. To incorporate prior knowledge of precedence relations among distributions, we impose stochastic order constraints and formulate the estimation task as a mixed-integer convex quadratic optimization problem. Experiments on both synthetic and real datasets show that the proposed method reduces the Jensen-Shannon divergence by 2.2% on average (up to 6.3%) when the sample size is small, while performing comparably to existing methods when sufficient data are available.

Problem

• discrete unimodal distributions
• stochastic order constraints
• simultaneous estimation
• search behavior analysis
• Jensen-Shannon divergence

Innovation

• stochastic order constraints
• unimodal distribution estimation
• mixed-integer convex optimization
• Jensen-Shannon divergence
• simultaneous estimation