Unifying ICL, SFT, KL-Regularized RL Through a Bayesian Lens

Supervised fine-tuning (SFT), few-shot in-context learning (ICL), KL-regularized RLHF/RLVR, and on-policy distillation are usually treated as distinct post-training paradigms. We develop a unified Bayesian perspective in which each is an instance of a two-step template: construct a (generalized) Bayes or Gibbs posterior from a reference model and a utility signal (log-likelihood, reward, or advantage), then approximate it by a forward-KL projection onto a parametric family, either in-weights (SFT/RL) or in-context (ICL). This yields a single chain of equivalences: few-shot ICL is an amortized projection onto the Bayes posterior predictive, and reward-weighted SFT, reward-weighted ICL, and advantage-weighted SFT are forward-KL projections of reward-induced Gibbs posteriors. The framework explains why supervised warm-up is practically unavoidable for importance-weighted projections, and interprets R1/o1-style reasoning models as combining test-time Bayesian search with training-time amortization. Matched-budget experiments on Qwen3 models corroborate the picture: operators that share their learning-signal granularity produce nearly identical updates when support is good and diverge when it degrades, and reward-weighted projection performs on par with standard baselines.