Differential Information Distribution: A Bayesian Perspective on Direct Preference Optimization

TL;DR

This paper offers a Bayesian framework for understanding Direct Preference Optimization (DPO), revealing how its reward structure, training dynamics, and downstream performance are driven by the concept of differential information, thus providing a theoretical foundation and practical insights.

Contribution

It introduces the Differential Information Distribution (DID) to formalize preference optimization from a Bayesian perspective, clarifying the rationale behind DPO's reward and its effects on training and performance.

Findings

DPO's log-ratio reward is justified by preferences encoding Differential Information.

Training dynamics follow a power-law relationship with DID.

High-entropy DID enhances open-ended instruction-following, low-entropy improves knowledge-intensive QA.

Abstract

Direct Preference Optimization (DPO) has been widely used for aligning language models with human preferences in a supervised manner. However, several key questions remain unresolved: the rationale behind its log-ratio reward, how the statistical structure of preference datasets shapes its training dynamics, and how those dynamics impact downstream capabilities. We approach these questions from a Bayesian perspective, interpreting the goal of preference optimization as learning the differential information required to update a reference policy into a target policy. To formalize this view, we introduce the Differential Information Distribution (DID), defined as the distribution over samples that carry the Bayesian evidence required to update policies. We introduce three complementary insights by viewing preference optimization through the DID. First, we find that DPO's log-ratio reward is uniquely justified when preferences encode the Differential Information needed to update a reference policy into the target policy. Second, we discuss how commonly observed training dynamics in DPO, including changes in log-likelihood and policy exploration, stem from a power-law DID relationship. Finally, we analyze how training dynamics influence downstream performance using the entropy of DID, a principled measure of uncertainty in the learned information. We observe that learning high-entropy DID improves open-ended instruction-following, while low-entropy DID benefits knowledge-intensive QA. Taken together, our results show that DPO's reward design, training dynamics, and downstream capabilities all emerge as natural consequences of learning Differential Information, offering both a principled theoretical foundation and practical guidance for preference-based alignment.