When Metropolis and Hastings Meet Bradley and Terry: Exact MCMC From Preference Voting
Abstract
Sampling from distributions conditioned on desired semantic properties is an emerging challenge in modern generative modeling. Metropolis-Hastings (MH) provides a principled route to conditional sampling, but requires access to exact pointwise target-density evaluations, which are not available in generative settings. Meanwhile, pairwise comparisons by humans or model "judge" are highly accessible and have proved valuable across diverse applications. We introduce Pref-MH, a general exact MH sampler for judge-induced conditional distributions using only stochastic binary pairwise comparisons. Our key observation is that the MH unnormalized density ratio matches the preference odds of the Bradley-Terry (BT) choice model. The central challenge is that while MH requires precise ratio computation, BT judges provide only sampled binary feedback. To this end, we develop a valid accept/reject rule whose resulting Markov chain provably converges to the target distribution. We further show that, for a fixed proposal kernel and budget, Pref-MH is optimal in the Peskun-Tierney sense among this class of exact reversible acceptance rules. Experiments on text generation and molecular design with LLM judges, as well as image generation with VLM judges, demonstrate that Pref-MH provides a practical and flexible approach to conditional sampling when comparative feedback is relatively easy to obtain.
Related
- Bradley-Terry model under general comparison graphs
- Alternative Strategies for Active Learning with Bayesian Bradley Terry Model
- Convergence of Independence Metropolis-Hastings Algorithms in High Dimensions: Iteration Lengths, Cutoff Profiles, and Diagnostic Unreliability
- Spectral Gap Inheritance for Metropolis-Hastings with Reversible Proposals
- Generalizing Metropolis-Hastings for DAG Sampling
BibTeX
@misc{smogorghevski2026when,
title = {{When Metropolis and Hastings Meet Bradley and Terry: Exact MCMC From Preference Voting}},
author = {Ariel Smogorghevski and Nir Rosenfeld and Yaniv Romano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00905},
eprint = {2609.00905},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00905},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - When Metropolis and Hastings Meet Bradley and Terry: Exact MCMC From Preference Voting
AU - Ariel Smogorghevski
AU - Nir Rosenfeld
AU - Yaniv Romano
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00905
UR - https://arxiv.org/abs/2609.00905
AB - Sampling from distributions conditioned on desired semantic properties is an emerging challenge in modern generative modeling. Metropolis-Hastings (MH) provides a principled route to conditional sampling, but requires access to exact pointwise target-density evaluations, which are not available in generative settings. Meanwhile, pairwise comparisons by humans or model "judge" are highly accessible and have proved valuable across diverse applications. We introduce Pref-MH, a general exact MH sampler for judge-induced conditional distributions using only stochastic binary pairwise comparisons. Our key observation is that the MH unnormalized density ratio matches the preference odds of the Bradley-Terry (BT) choice model. The central challenge is that while MH requires precise ratio computation, BT judges provide only sampled binary feedback. To this end, we develop a valid accept/reject rule whose resulting Markov chain provably converges to the target distribution. We further show that, for a fixed proposal kernel and budget, Pref-MH is optimal in the Peskun-Tierney sense among this class of exact reversible acceptance rules. Experiments on text generation and molecular design with LLM judges, as well as image generation with VLM judges, demonstrate that Pref-MH provides a practical and flexible approach to conditional sampling when comparative feedback is relatively easy to obtain.
ER -