This research considers a scalable inference for spatial data modeled through Gaussian intrinsic conditional autoregressive (ICAR) structures. The classical estimation method, restricted maximum likelihood (REML), requires repeated inversion and factorization of large, sparse precision matrices, which makes this computation costly. To sort this problem out, we propose a variational restricted maximum likelihood (VREML) framework that approximates the intractable marginal likelihood using a Gaussian variational distribution. By constructing an evidence lower bound (ELBO) on the restricted likelihood, we derive a computationally efficient coordinate-ascent algorithm for jointly estimating the spatial random effects and variance components. In this article, we theoretically establish the monotone convergence of ELBO and mathematically exhibit that the variational family is exact under Gaussian ICAR settings, which is an indication of nullifying approximation error at the posterior level. We empirically establish the supremacy of our VREML over MLE and INLA.

翻译：本研究探讨了通过高斯内在条件自回归（ICAR）结构建模的空间数据的可扩展推断问题。经典估计方法——约束最大似然（REML）——需要反复对大型稀疏精度矩阵求逆及分解，导致计算成本高昂。为解决此问题，我们提出变分约束最大似然（VREML）框架，利用高斯变分分布近似难以处理的边际似然。通过构建约束似然的证据下界（ELBO），我们推导出一种计算高效的坐标上升算法，用于联合估计空间随机效应及方差分量。本文从理论上证明了ELBO的单调收敛性，并从数学上揭示了变分族在高斯ICAR设定下的精确性，这标志着后验层面的近似误差可被归零。我们通过实证验证了VREML相对于MLE和INLA的优越性。