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.

We perform experiments on high-dimensional Gaussians, Lotka-V olterra dynamics, and a Pickover attractor.

We develop EigenVI, an eigenvalue-based approach for black-box variational inference (BBVI).

In variational inference, the benefits of Bayesian models rely on accurately capturing the true posterior distribution. We propose using neural samplers that specify implicit distributions, which are well-suited for approximating complex multimodal and correlated posteriors in high-dimensional spaces.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Shared variability is separated into terms that express condition dependence, as well as trial-to-trial variation in trajectories.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

By now Bayesian methods are routinely used in practice for solving inverse problems.