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.
Large language models (LLMs) are trained on corpora that contain expressions of human judgment about films, books, music, and more. Yet whether LLMs systematically reproduce evaluative hierarchies remains unclear. Prior research on cultural bias in LLMs suggests competing expectations: models may mirror the popularity signals of internet texts, or may reproduce forms of prestige embedded in critical discourse. We probe this question through a study of film evaluations with eight models from four families (Anthropic, OpenAI, Alibaba, and Mistral), using a 200-film benchmark partitioned into critically acclaimed, commercially successful, and dual-legitimacy (critical acclaim + commercial success) films. Across 20,000 pairwise forced-choice comparisons per model analyzed with Bradley--Terry estimation, we observe a consistent critical acclaim orientation with all models: critically acclaimed yet commercially obscure films are selected over commercially successful yet critically unrecognized ones. This pattern grows with model scale within each family. In addition, nested OLS regression analyses show that evaluative orientation, public visibility, and popular reception distinctly help explain preferences. Adjusting for public visibility reverses the models' preference for dual-legitimacy films over critical acclaim-only films, while additionally accounting for popular reception attenuates much of the disadvantage of films with commercial success only. Finally, evaluative and recommendation-oriented prompt framings produce divergent rankings, suggesting that critical acclaim orientation may manifest indirectly in real-world LLM deployments.
In this paper, we study prediction problems for paired comparison data, for example, predicting the win probability between two unmatched players and ranking all the players according to the order of their strengths by using win probability data between two matched players. Paired comparison data are typically analyzed using Bradley-Terry and Thurstone-Mosteller models. These models predict the win probability by transforming the difference between learned rate parameters, which represent players'\;strengths, with a pre-specified inverse link function, and employ the order of learned rate parameters for player ranking. However, these models may suffer from model misspecification owing to the selection of a fixed inverse link function. Therefore, in this study, we propose to learn the rate parameters by a (sub-)gradient method and the inverse link function by an isotonic regression technique alternately. The proposed model guarantees monotonic improvement in training error, and is likely to yield an exact tie when the available data is insufficient to establish a strict ranking. We also verified that the proposed model could improve the win probability prediction and ranking performance through numerical experiments with synthetic data and real-world data of football Premier League, baseball MLB, and tennis ATP tour.
Zermelo's algorithm is a classical method for computing the maximum likelihood estimator in the Bradley--Terry (BT) model, but its convergence can be slow in practice. To accelerate computation, Newman introduced a family of Zermelo-type fixed-point iterations parameterized by $α$, with Zermelo's algorithm recovered at $α=1$. Empirical evidence suggests that the choice $α=0$ often converges substantially faster, making it a promising alternative, yet the mechanism underlying this acceleration remains elusive. This paper provides theoretical insight into this phenomenon through a systematic local convergence analysis. We derive closed-form expressions for local convergence factors under synchronous and asynchronous updates and analyze their dependence on $α$ via spectral analysis of the associated Jacobian matrices. For synchronous updates, we show that the algorithm may fail to converge when $α<1$, and its local convergence factor is quasi-convex in $α$ under the population BT model. In contrast, asynchronous updates are always locally convergent, and their local convergence factor is provably monotonically increasing in $α$ under the population BT model of consistently ordered bipartite comparison graphs, establishing the optimality of $α=0$ in this setting. We further establish asymptotic approximation results for the population convergence factors under the BT model, justifying their practical relevance. Numerical experiments on synthetic and real-world datasets confirm the theory. Our analysis complements existing convergence results and shows that the acceleration of $α=0$ arises not only from the parameter choice but, more importantly, from the use of asynchronous updates.