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Transfer learning for probabilistic regression remains underexplored. This work closes this gap by introducing NIAQUE, Neural Interpretable Any-Quantile Estimation, a new model designed for transfer learning in probabilistic regression through permutation invariance. We demonstrate that pre-training NIAQUE directly on diverse downstream regression datasets and fine-tuning it on a specific target dataset enhances performance on individual regression tasks, showcasing the positive impact of probabilistic transfer learning. Furthermore, we highlight the effectiveness of NIAQUE in Kaggle competitions against strong baselines involving tree-based models and recent neural foundation models TabPFN and TabDPT. The findings highlight NIAQUE's efficacy as a robust and scalable framework for probabilistic regression, leveraging transfer learning to enhance predictive performance.