Constrained Bayesian Reconstruction of Latent Public Preferences Under Dynamic Elimination Rules: An Uncertainty-Aware Expert–Public Fusion Framework

Abstract

Expert–public selection systems often reveal expert scores, institutional rules, and discrete elimination outcomes while concealing public-vote information. This paper develops a constrained Bayesian inverse-inference framework for inferring rule-compatible latent public-preference distributions. Public preferences are represented as simplex-valued latent variables, and rank-based, percentage-based, and judges’ save mechanisms are encoded through rule-induced likelihood constraints. We formally characterize the rule-compatible identified-set geometry: the percentage-rule set is a compact convex polytope, whereas the rank-based and judges’ save sets are finite unions of polyhedral rank cells. Prior-free sharp coordinate bounds are obtained by exact linear programming for all 170 percentage-rule weeks and by exact weak-order enumeration in representative rank and judges’ save cases. Posterior distributions are approximated using a mixed Metropolis–Hastings sampler whose transition kernel preserves the target posterior; the positive-probability global Dirichlet component provides irreducibility across separated positive-posterior-mass regions. In the primary 1800-case factorial synthetic study spanning two truth-generating mechanisms, three contestant-set sizes, and three institutional rules, marginal 95% credible-interval coverage was similar for the rule-constrained posterior (0.9400–0.9625) and a newly added prior-only benchmark (0.9475–0.9617), indicating that near-nominal coverage primarily reflects interval calibration rather than recovery accuracy. Conditioning on the observed elimination reduced the overall mean total-variation distance from 0.4613 to 0.4353 and increased the mean Spearman correlation from approximately zero (−0.0206) to 0.2147, although gains in Top-1 and Top-3 identification were not uniform under logistic-normal misspecification. A 54-case validation showed close agreement between vectorized rejection sampling and mixed Metropolis–Hastings sampling. In an empirical application to 34 seasons of Dancing with the Stars, multi-chain diagnostics provided no material evidence of non-convergence; posterior mean shares were stable under moderate specification changes, whereas rankings and uncertainty widths were more sensitive. Across 224 rule-constrained weeks, the Logistic Dynamic Weighting System agreed with percentage-based decisions in 86.66% of same-draw comparisons. The framework is therefore uncertainty-aware and institutionally transparent, rather than a method for exact vote recovery or a universally superior aggregation rule.