用概率程序生成训练数据，让大模型学会处理不确定的归纳推理。

Post-training Large Language Models (LLMs) for reasoning typically focuses on deductive tasks such as mathematics and coding where correctness is verifiable. Yet, many real-world reasoning problems are inductive: agents must infer uncertain beliefs from sparse, ambiguous observations. There are challenges to using standard fine-tuning methods for inductive reasoning, including difficulties in curating large-scale, high-quality labeled datasets and in handling targets that are inherently distributional. In this work, we introduce a novel approach, called Program-based Posterior Training (PPT), to address these limitations: we use an LLM to generate diverse open-world scenarios as probabilistic programs, run probabilistic inference to produce distributional target responses to queries, and then fine-tune on these probabilistic soft labels. Using this approach, we fine-tune LLMs on 10,000 programmatically generated scenarios and evaluate on held-out motifs, human-labeled judgments, and external benchmarks. Overall, PPT substantially improves estimation accuracy on held-out inductive tasks, increases alignment with human judgments, and transfers to external benchmarks for estimation and calibration. Additionally, the gains in raw calibration are not subsumed by post-hoc temperature scaling, showing that the models have more deeply internalized uncertainty compared to output rescaling. Together, these results suggest that probabilistic-program-mediated fine-tuning is a promising approach for post-training LLMs to reliably perform approximate inductive inference.