The integrated nested Laplace approximations (INLA) method has become a widely utilized tool for researchers and practitioners seeking to perform approximate Bayesian inference across various fields of application. To address the growing demand for incorporating more complex models and enhancing the method's capabilities, this paper introduces a novel framework that leverages dense matrices for performing approximate Bayesian inference based on INLA across multiple computing nodes using HPC. When dealing with non-sparse precision or covariance matrices, this new approach scales better compared to the current INLA method, capitalizing on the computational power offered by multiprocessors in shared and distributed memory architectures available in contemporary computing resources and specialized dense matrix algebra. To validate the efficacy of this approach, we conduct a simulation study then apply it to analyze cancer mortality data in Spain, employing a three-way spatio-temporal interaction model.

翻译：集成嵌套拉普拉斯近似（INLA）方法已成为各应用领域研究人员和实践者广泛使用的近似贝叶斯推断工具。为应对日益增长的复杂模型需求并提升方法性能，本文提出了一种基于稠密矩阵的新型框架，利用高性能计算在多个计算节点上执行基于INLA的近似贝叶斯推断。当处理非稀疏精度矩阵或协方差矩阵时，该新方法相比现有INLA方法具有更好的扩展性，其充分利用了当代计算资源中共享与分布式内存架构的多处理器计算能力以及专门化稠密矩阵代数。为验证该方法的有效性，我们开展了仿真研究，并采用三向时空交互模型将其应用于西班牙癌症死亡率数据分析。