BayesPrompt: human readable prompts that make sense

Prompt optimization adapts large language models (LLMs) without updating model parameters, but many automatic prompt optimizers remain heuristic search procedures over candidate instructions. This paper studies prompt optimization as Bayesian posterior sampling over discrete prompt tokens. We define a posterior distribution by combining a task likelihood term, which rewards prompts that explain input-output examples, with a language-model prior, which favors fluent instructions. This converts prompt optimization into an energy-based posterior sampling problem, for which gradients can be used to guide discrete Markov chain Monte Carlo (MCMC) proposals over vocabulary tokens. We refer to our framework as BayesPO, short for Bayesian Prompt Optimization. In this paper, BayesPO is instantiated with Markov chain Monte Carlo: it uses a Metropolis-Hastings corrected Gibbs-with-Langevin (GwL) proposal and integrates parallel tempering for global exploration of rugged LLM-induced energy landscapes. The concrete sampler further adapts the GwL sampler to the practical constraints of non-weight-tied LLM embeddings. Experiments with Qwen2.5 models show that the sampler discovers semantically meaningful prompts on diagnostic tasks, that parallel tempering helps escape a local optimum in a poetry completion task, and that post-optimizing APE prompts on 24 instruction-induction subtasks improves average accuracy from 60.04% to 63.23%. The study also reveals two main limitations: energy minimization may overfit small optimization sets, and the current sampler remains computationally expensive. These findings position Bayesian prompt sampling as a principled post-optimization tool and point to a promising direction for probabilistic prompt optimization.

In this paper, we define the quantity of prompting complexity: for a fixed instruction-tuned language model, what is the shortest plausible prompt that makes deterministic decoding produce a target text? It is an LM-relative analogue of resource-bounded Kolmogorov complexity: the prompt is a program, the model interface is the interpreter, and information omitted from the prompt is supplied by the model's weights, training distribution, tokenizer, template, and decoding rule. Unlike classical Kolmogorov complexity, this measure is intentionally non-universal. In the finite-context setting it is computable by enumeration, but there is no model-independent invariance theorem; the same text may be cheap for one model and inaccessible or expensive for another. To keep the search space aligned with prompt engineering, we restrict programs to plausible human-readable texts rather than arbitrary token strings. We extend the exact definition to soft prompting complexity for approximate outputs, yielding a lossy notion of model-relative text compression and a formal target for prompt optimization. We also define prompting distance by comparing shortest generating prompts, and behavioral prompting complexity for reaching any output satisfying a specification. Based on these formulations, we define a research agenda for empirically studying which texts and behaviors are accessible from short plausible prompts under a fixed LM interface.

System prompts are a central control mechanism in modern AI systems, shaping behavior across conversations, tasks, and user populations. Yet they are difficult to tune when feedback is available only as aggregate metrics rather than per-example labels, failures, or critiques. We study this aggregate feedback setting as sample-constrained black-box optimization over discrete, variable-length text. We introduce ReElicit, a Bayesian optimization framework based on embedding by elicitation. Given a task description, previously evaluated prompts, and scalar scores, an LLM elicits a compact, interpretable feature space and maps prompts into it. Leveraging a probabilistic Gaussian process surrogate, an acquisition function then selects target feature vectors, which the LLM realizes and refines into deployable system prompts. Re-eliciting the feature space as new evaluations arrive lets the representation adapt to the observed prompt-score history. We evaluate the setting using offline benchmark accuracy as a controlled aggregate proxy: the optimizer observes one scalar score per prompt and no per-example labels, errors, or critiques. Across ten system prompt optimization tasks with a 30 total evaluation budget, ReElicit achieves the strongest aggregate performance profile among representative aggregate-only prompt-optimization baselines. These results suggest that LLMs can serve as adaptive semantic representation builders, not only prompt generators, for Bayesian optimization over natural-language artifacts.