Scalable Sparse Autoregressive Variational Inference for Non-Gaussian Spatial Fields
Authors: Paul F. V. Wiemann, Matthias Katzfuss
Organizations: Department of Statistics, The Ohio State University, Columbus OH 43210, USA · Department of Statistics, University of Wisconsin–Madison, Madison WI 53706, USA
Latent field models are essential for spatial statistics, but the posterior distribution of the latent field, even with a Gaussian process (GP) prior, is analytically intractable whenever the likelihood is non-Gaussian. Existing approximations are either too slow or too restrictive for large non-Gaussian datasets. We present a scalable variational inference framework that mirrors the sparse nearest-neighbor autoregressive structure of the prior in the variational family, uses normalizing-flow conditionals for expressiveness, and amortizes all variational parameters via a single shared neural conditioner. Doubly stochastic optimization with reduced ancestor sets achieves polylogarithmic complexity per iteration.
Figures & tables
Figure 1: Maximin ordering of n=512 points. For selected indices i (black dot), we show the conditioning set Ci (blue), full ancestor set Ai (green), and reduced ancestor set Ri (orange). Clearly for large i , ∣Ai∣>∣Ri∣ .
Figure 2: KL divergence to the exact posterior distribution vs. ρ for noisy GP regression ( n=50 ), using full and reduced ancestor sets. Accuracy improves monotonically with ρ .
Figure 3: GP binary classification on the unit interval. Our two-piece-normal NN-AR family (middle) captures posterior skewness and achieves higher a ELBO than the Gaussian baseline (left). The right panel compares the estimated densities for f11 .
Figure 4: Left to right: Observed data, posterior mean (non-amortized and amortized), posterior standard deviation (non-amortized and amortized), posterior mean (larger dataset, amortized).