提出新方法加速空间数据的高效推断，计算更快且无近似误差。
Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data
- 用变分推断近似难解的边际似然，构建可优化的下界目标
- 算法在真实数据上比MLE和INLA快3~5倍，精度相当
- 理论证明在高斯ICAR模型下无近似误差，适合大规模空间建模
本文研究通过高斯固有条件自回归（Gaussian ICAR）结构建模的空间数据的可扩展推断问题。经典受限最大似然（REML）方法需反复对大型稀疏精度矩阵进行求逆和分解，计算成本高昂。为此，我们提出变分受限最大似然（VREML）框架，利用高斯变分分布近似不可解析的边际似然。通过构建受限似然的证据下界（ELBO），推导出一种计算高效的坐标上升算法，联合估计空间随机效应与方差成分。本文理论上证明了ELBO的单调收敛性，并数学上展示了在高斯ICAR设定下，变分族是精确的，意味着后验层面可消除近似误差。实证表明，VREML在性能上显著优于MLE和INLA。
原文摘要 · Abstract (English)
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.
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