Prompt optimisation is how you adapt a language model without touching its weights: keep rewriting the instruction until the outputs improve. Most automatic optimisers do this as heuristic search, trying candidate instructions and keeping whatever scores well, which works but offers no account of what is actually being searched for.

This paper reframes the task as Bayesian posterior sampling over discrete prompt tokens. The posterior combines two terms: a task likelihood that rewards prompts explaining the input-output examples, and a language-model prior that favours instructions which read fluently. The second term is doing real work, because it pushes back against the garbled strings that pure score-chasing tends to discover.

The reframing turns a bag of heuristics into energy-based posterior sampling, which means the goal is a distribution over good prompts rather than a single lucky winner. That is a more honest description of the problem, since there was never one uniquely correct instruction to begin with.