When Metropolis and Hastings Meet Bradley and Terry: Exact MCMC From Preference Voting

Conditional generators provide a natural tool for controllable generation, including settings where the desired condition is a new composition of observed attributes or experimental factors. In many applications, especially in scientific domains, such models are attractive to explore conditions for which real samples are rare, expensive, or not yet observed. However, this creates a circularity for evaluation: standard conditional quality metrics require a reference target distribution, but in the extrapolative regime that distribution is unavailable by definition. We address this problem with a post-hoc, per-sample trust score for assessing conditional samples using only the training distribution. The score combines two estimable quantities: global realism, measuring compatibility with the real data manifold, and attribute-wise faithfulness, measuring whether a sample is closer to the requested attributes than to plausible alternatives. We show that the score can recover meaningful comparisons across extrapolated generations, under a mild coverage condition on the observed attributes. These comparisons enable effective filtering, ranking, and abstention of generations and can be used directly on off-the-shelf pretrained models. In biological imaging, selected samples preserve real morphological structure better and improve downstream predictive performance, while similar gains are observed on controlled vision benchmarks. Finally, we show how the score can be applied during generation, enabling abstention before full decoding. Code is available at https://github.com/berkerdemirel/faithful-cond-gen.

We propose Markov Chain from Human Feedback (MCHF), an elementary approach for aligning generative models from pairwise human preferences. Unlike Reinforcement Learning from Human Feedback (RLHF), which reduces comparisons to a scalar reward, and Nash Learning from Human Feedback (NLHF), which preserves pairwise utilities through a KL-regularized minimax optimization, MCHF uses pairwise preferences directly to define a transition mechanism over model outputs. Given a pairwise utility $U(x,y)$, which quantifies human preference for $y$ over $x$, and a reference probability distribution $μ_{\mathsf{ref}}$, we define a Markov kernel $\mathsf{P}(x, dy)\propto \exp(U(x,y))μ_{\mathsf{ref}}(dy)$, and take the Markov chain starting from $μ_{\mathsf{ref}}$ as an iterative alignment procedure. We show that MCHF converges geometrically fast to the stationary distribution, with a convergence rate governed by the seminorm $\|U\|_\oplus=\inf_{g,f\in L^\infty(μ_{\mathsf{ref}})}\|U-g\oplus f\|_\infty$, which quantifies the non-transitive structure of the pairwise utility. We further show that a mirror-descent algorithm for NLHF satisfies an analogous structure-adaptive convergence guarantee. Finally, through a perturbation analysis, we prove that when $\|U\|_\oplus$ is small, MCHF and NLHF agree up to first order around an RLHF solution, which yields a unified view of reward-based, game-theoretic, and Markovian approaches to alignment. In particular, for two natural algorithms that converge to the MCHF/NLHF equilibria, we show that the first step of MCHF and NLHF recovers the RLHF solution based on the column-sum reward $\hat{f}(y)=\int μ_{\mathsf{ref}}(dx) U(x, y)$, and starting from the second iteration, both algorithms incorporate the same linear functional of the residual $U-(-\hat f)\oplus \hat f$, which captures the non-transitive structure of the pairwise utility $U$.

Conditional generative models have emerged as powerful tools for sampling from target conditional distributions, driving substantial advances across a wide range of scientific and applied domains. As these models proliferate, practitioners often face multiple plausible generators whose performance can vary with the task, data, or input condition. We propose an optimal model averaging framework for conditional generative models, allowing candidate generators to be combined even when they are accessible only through conditional samples without tractable densities. Specifically, we use a sample-based maximum mean discrepancy between conditional distributions, which first leads to a static model averaging method, StaticMA, assigning fixed weights to different candidates. In addition, we develop MoEMA (mixture-of-experts model averaging), an input-adaptive method that parameterizes covariate-dependent weights through a softmax neural-network gate. We establish in-sample and out-of-sample asymptotic optimality for the proposed methods, together with consistency of the estimated adaptive weight function under regularity conditions. The framework applies directly to Euclidean responses and extends to unstructured data by combining our formulation with fixed representation maps. Across a broad set of simulations and real-data studies spanning tabular, image, and text modalities, MoEMA generally improves over competing baselines, demonstrating the effectiveness of our proposed methods.