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Ostrom-Weighted Bootstrap: A Theoretically Optimal and Provably Complete Framework for Hierarchical Imputation in Multi-Agent Systems

Hirofumi Wakimoto

TL;DR

OWB is the first resampling-based method that simultaneously achieves exact BLUE optimality, conditional Bayesian posterior mean interpretation, empirical Bayes shrinkage of precision parameters, asymptotic efficiency via FGLS, consistent weighted bootstrap inference, and provable zero-NaN completion under minimal data assumptions.

Abstract

We study the statistical properties of the \emph{Ostrom-Weighted Bootstrap} (OWB), a hierarchical, variance-aware resampling scheme for imputing missing values and estimating archetypes in multi-agent voting data. At Level~1, under mild linear model assumptions, the \emph{ideal} OWB estimator -- with known persona-level (agent-level) variances -- is shown to be the Gauss--Markov best linear unbiased estimator (BLUE) and to strictly dominate uniform weighting whenever persona variances differ. At Level~2, within a canonical hierarchical normal model, the ideal OWB coincides with the conditional Bayesian posterior mean of the archetype. We then analyze the \emph{feasible} OWB, which replaces unknown variances with hierarchically pooled empirical estimates, and show that it can be interpreted as both a feasible generalized least-squares (FGLS) and an empirical-Bayes shrinkage estimator with asymptotically valid weighted bootstrap confidence intervals under mild regularity conditions. Finally, we establish a Zero-NaN Guarantee: as long as each petal has at least one finite observation, the OWB imputation algorithm produces strictly NaN-free completed data using only explicit, non-uniform bootstrap weights and never resorting to uniform sampling or numerical zero-filling. To our knowledge, OWB is the first resampling-based method that simultaneously achieves exact BLUE optimality, conditional Bayesian posterior mean interpretation, empirical Bayes shrinkage of precision parameters, asymptotic efficiency via FGLS, consistent weighted bootstrap inference, and provable zero-NaN completion under minimal data assumptions.

Ostrom-Weighted Bootstrap: A Theoretically Optimal and Provably Complete Framework for Hierarchical Imputation in Multi-Agent Systems

TL;DR

OWB is the first resampling-based method that simultaneously achieves exact BLUE optimality, conditional Bayesian posterior mean interpretation, empirical Bayes shrinkage of precision parameters, asymptotic efficiency via FGLS, consistent weighted bootstrap inference, and provable zero-NaN completion under minimal data assumptions.

Abstract

We study the statistical properties of the \emph{Ostrom-Weighted Bootstrap} (OWB), a hierarchical, variance-aware resampling scheme for imputing missing values and estimating archetypes in multi-agent voting data. At Level~1, under mild linear model assumptions, the \emph{ideal} OWB estimator -- with known persona-level (agent-level) variances -- is shown to be the Gauss--Markov best linear unbiased estimator (BLUE) and to strictly dominate uniform weighting whenever persona variances differ. At Level~2, within a canonical hierarchical normal model, the ideal OWB coincides with the conditional Bayesian posterior mean of the archetype. We then analyze the \emph{feasible} OWB, which replaces unknown variances with hierarchically pooled empirical estimates, and show that it can be interpreted as both a feasible generalized least-squares (FGLS) and an empirical-Bayes shrinkage estimator with asymptotically valid weighted bootstrap confidence intervals under mild regularity conditions. Finally, we establish a Zero-NaN Guarantee: as long as each petal has at least one finite observation, the OWB imputation algorithm produces strictly NaN-free completed data using only explicit, non-uniform bootstrap weights and never resorting to uniform sampling or numerical zero-filling. To our knowledge, OWB is the first resampling-based method that simultaneously achieves exact BLUE optimality, conditional Bayesian posterior mean interpretation, empirical Bayes shrinkage of precision parameters, asymptotic efficiency via FGLS, consistent weighted bootstrap inference, and provable zero-NaN completion under minimal data assumptions.
Paper Structure (7 sections, 5 theorems, 15 equations)

This paper contains 7 sections, 5 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Naming and conceptual motivation.
  3. Related Work
  4. Model Setup and Notation
  5. Level 1 Theorems: Finite-Sample Linear Optimality
  6. Level 2 Theorems: Hierarchical Normal Model
  7. Implementation Proposition: Zero-NaN Guarantee

Key Result

Theorem 1

Fix a petal $j$ and consider linear estimators of the form Under Assumption ass:level1, the ideal OWB estimator where we set $v_{\mathrm{eff},p} = v_p$, it satisfies:

Theorems & Definitions (11)

  • Theorem 1: Level 1: GLS Optimality and Dominance over Uniform Weights
  • proof : Proof sketch
  • Theorem 2: Level 1: Hierarchical Pooling and Shrinkage
  • Remark 1
  • Theorem 3: Level 2: Ideal OWB as BLUE and Conditional Posterior Mean
  • proof : Proof sketch
  • Theorem 4: Level 2: Feasible OWB as FGLS + EB + Weighted Bootstrap
  • Remark 2: On James--Stein admissibility
  • Remark 3: On the weight lower-bound assumption
  • Proposition 1: Zero-NaN Guarantee
  • ...and 1 more