The Jacobi prior offers an alternative Bayesian framework, designed to achieve superior computational efficiency without compromising predictive performance. Compared to widely used methods such as Lasso, Ridge, Elastic Net, uniLasso, the MCMC-based Horseshoe prior, and non-Bayesian machine learning methods including Support Vector Machines (SVM), Random Forests, and Extreme Gradient Boosting (XGBoost), the Jacobi prior achieves competitive or better accuracy with significantly reduced computational cost. The method is well suited to distributed computing environments, as it naturally accommodates partitioned data across multiple servers. We propose a parallelisable Monte Carlo algorithm to quantify the uncertainty in the estimated coefficients. We establish that the Jacobi estimator is asymptotically close to, and asymptotically equivalent to, the posterior mode under the Jacobi prior. To demonstrate its practical utility, we conduct a comprehensive simulation study comprising seven experiments focused on statistical consistency, prediction accuracy, scalability, sensitivity analysis and robustness study. We further present three real-data applications multi-class classification of stars, quasars, and galaxies using Sloan Digital Sky Survey data, and spinal degeneration classification using sagittal MRI scans from the RSNA 2024 Lumbar Spine Degenerative Classification Challenge. In the spine classification task, we extract last-layer features from a fine-tuned ResNet-50 model and evaluate multiple classifiers, including Jacobi-Multinomial logit regression, SVM, and Random Forest. All code and datasets used in this paper are available at: https://github.com/sourish-cmi/Jacobi-Prior/

翻译：雅可比先验提供了一种替代的贝叶斯框架，旨在实现卓越的计算效率，同时不牺牲预测性能。与广泛使用的方法（如Lasso、Ridge、Elastic Net、uniLasso、基于MCMC的Horseshoe先验）以及非贝叶斯机器学习方法（包括支持向量机（SVM）、随机森林和极限梯度提升（XGBoost））相比，雅可比先验以显著降低的计算成本实现了相当或更优的准确性。该方法非常适合分布式计算环境，因为它天然适应跨多个服务器的分区数据。我们提出了一种可并行的蒙特卡洛算法来量化估计系数的不确定性。我们证明了雅可比估计量在渐近意义上接近，并且在渐近等价于雅可比先验下的后验众数。为了展示其实用性，我们进行了一项全面的模拟研究，包含七个实验，重点关注统计一致性、预测准确性、可扩展性、敏感性分析和鲁棒性研究。我们进一步展示了三个真实数据应用：使用斯隆数字巡天数据对恒星、类星体和星系进行多类别分类，以及使用来自RSNA 2024腰椎退行性变分类挑战赛的矢状面MRI扫描进行脊柱退行性变分类。在脊柱分类任务中，我们从微调后的ResNet-50模型中提取最后一层特征，并评估了多种分类器，包括雅可比-多项对数回归、SVM和随机森林。本文使用的所有代码和数据集均可在以下网址获取：https://github.com/sourish-cmi/Jacobi-Prior/