Fan Vote Share Estimation Based on Constrained Inversion and Bayesian Validation

Abstract

This study develops a fan vote share estimation model for competition settings in which weekly audience support is not directly observable. The model uses known judge scores, elimination outcomes, and rule mechanisms to infer contestant-level fan vote shares through constrained nonlinear optimization. The framework defines active contestants by elimination week, aggregates multi-judge scores, standardizes judge-score shares, and combines judge scores with estimated fan vote shares under percentage-based scoring rules. To address the non-uniqueness of inverse estimation, the model introduces temporal smoothness and maximum entropy regularization, while hard elimination constraints ensure that inferred results reproduce observed eliminations. The SLSQP algorithm is used for sequential season-week optimization, and Bootstrap perturbation provides confidence intervals and certainty indices. Verification results include champion prediction, cross-validation, residual analysis, and Bayesian latent popularity validation. The model explains cases in which low judge scores coexist with high fan support and shows strong consistency between constrained inversion and Bayesian posterior estimates.