# The Missing Coefficients: Bayesian Pairwise Merging for Model Personalization

arXiv: https://arxiv.org/abs/2609.39055 · HTML: https://mlsanity.com/paper/2609.39055/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 67 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 30 of 36)

- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Microsoft COCO Captions: Data Collection and Evaluation Server](https://mlsanity.com/paper/1504.00325/) (2015)
- [Hierarchical Neural Story Generation](https://mlsanity.com/paper/1805.04833/) (2018)
- [Fine-Grained Human Feedback Gives Better Rewards for Language Model Training](https://mlsanity.com/paper/2306.01693/) (2023)
- [Rewarded soups: towards Pareto-optimal alignment by interpolating weights fine-tuned on diverse rewards](https://mlsanity.com/paper/2306.04488/) (2023)
- [GDPO: Group reward-Decoupled Normalization Policy Optimization for Multi-reward RL Optimization](https://mlsanity.com/paper/2601.05242/) (2026)
- [Rewards-in-Context: Multi-objective Alignment of Foundation Models with Dynamic Preference Adjustment](https://mlsanity.com/paper/2402.10207/) (2024)
- [Personalizing Reinforcement Learning from Human Feedback with Variational Preference Learning](https://mlsanity.com/paper/2408.10075/) (2024)
- [Arithmetic Control of LLMs for Diverse User Preferences: Directional Preference Alignment with Multi-Objective Rewards](https://mlsanity.com/paper/2402.18571/) (2024)
- [GREEN: Generative Radiology Report Evaluation and Error Notation](https://mlsanity.com/paper/2405.03595/) (2024)
- [Beyond One-Preference-Fits-All Alignment: Multi-Objective Direct Preference Optimization](https://mlsanity.com/paper/2310.03708/) (2023)
- [BoNBoN Alignment for Large Language Models and the Sweetness of Best-of-n Sampling](https://mlsanity.com/paper/2406.00832/) (2024)
- [Decoding-Time Language Model Alignment with Multiple Objectives](https://mlsanity.com/paper/2406.18853/) (2024)
- [Batch Active Preference-Based Learning of Reward Functions](https://mlsanity.com/paper/1810.04303/) (2018)
- [Panacea: Pareto Alignment via Preference Adaptation for LLMs](https://mlsanity.com/paper/2402.02030/) (2024)
- [Of Human Criteria and Automatic Metrics: A Benchmark of the Evaluation of Story Generation](https://mlsanity.com/paper/2208.11646/) (2022)
- [Asking Easy Questions: A User-Friendly Approach to Active Reward Learning](https://mlsanity.com/paper/1910.04365/) (2019)
- [Learning to Summarize Radiology Findings](https://mlsanity.com/paper/1809.04698/) (2018)
- [Conditional Language Policy: A General Framework for Steerable Multi-Objective Finetuning](https://mlsanity.com/paper/2407.15762/) (2024)
- [Inverse Constitutional AI: Compressing Preferences into Principles](https://mlsanity.com/paper/2406.06560/) (2024)
- [A Survey on Personalized Alignment -- The Missing Piece for Large Language Models in Real-World Applications](https://mlsanity.com/paper/2503.17003/) (2025)
- [LoRe: Personalizing LLMs via Low-Rank Reward Modeling](https://mlsanity.com/paper/2504.14439/) (2025)
- [Language Model Personalization via Reward Factorization](https://mlsanity.com/paper/2503.06358/) (2025)
- [PARM: Multi-Objective Test-Time Alignment via Preference-Aware Autoregressive Reward Model](https://mlsanity.com/paper/2505.06274/) (2025)
- [Promptable Behaviors: Personalizing Multi-Objective Rewards from Human Preferences](https://mlsanity.com/paper/2312.09337/) (2023)
- [Autorubric: A Unifying Framework for Rubric-Based LLM Evaluation on Non-Verifiable Tasks](https://mlsanity.com/paper/2603.00077/) (2026)
- [MAP: Low-compute Model Merging with Amortized Pareto Fronts via Quadratic Approximation](https://mlsanity.com/paper/2406.07529/) (2024)

## Abstract (arXiv)

How can we personalize a shared expert library from a user's pairwise choices? Prior work can realize different reward trade-offs by merging reward-specialized experts, given a vector of trade-off weights. In practice, users can more naturally choose between outputs than specify numerical weights. The challenge is therefore to turn these choices into the coefficients required for merging, while accounting for ambiguity when feedback is limited. Our key idea is to treat the unknown reward weights as latent variables: infer a posterior over them from pairwise choices and reward-score differences, and use its mean directly as the merge coefficients. We instantiate this idea as Bayesian Pairwise Merging (BPM), whose posterior also characterizes which reward trade-offs remain plausible given the feedback. We evaluate BPM on radiology summarization, image captioning, and story generation, spanning text-to-text and image-to-text generation. With 100 feedback per simulated persona, BPM achieves macro decided win rates of 91.7%, 77.1%, and 64.3% against uniform merge. For six pairs of simulated personas, each prefers the model fitted to its own feedback, a pattern also observed in a human proof-of-concept. In simulations under BPM's model and prior, its nominal 90% intervals for temperature-scaled reward weights achieve task-averaged marginal coverage of 88.9% and 89.2% with only 10 and 25 comparisons, respectively. BPM thus enables personalization from pairwise feedback without per-user policy training, while characterizing the coefficient ambiguity left by limited feedback.