  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.
