Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data

Debjoy Thakur

stat.MLApr 8, 2026

Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data

Authors: Debjoy Thakur

Organizations: Washington University in St. Louis, USA · Ahmedabad University, India

Abstract

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.

Figures & tables

Figure 1: Mean Kullback–Leibler divergence between the variational posterior,

q(\bm{u})

and the true posterior,

g_{\bm{\theta}}(\bm{u}|\bm{Y})

across different sample sizes.

Method	$n$	Avg. MSPE	Avg. Posterior $\bm{u}$ Mean MSE	Avg. Posterior $\bm{u}$ Variance MSE	Avg. $\bm{\beta}$ MSE
INLA	25	0.989	0.468	0.447	0.414
INLA	64	1.522	0.708	0.482	0.183
INLA	100	1.283	0.846	0.498	0.096
INLA	400	1.322	0.437	0.049	0.029
INLA	625	1.139	0.527	0.001	0.030
INLA	900	1.306	0.587	0.038	0.051

Table 1: Simulation performance comparison across methods and sample sizes.

Figure 2: Average RMSE comparison for

\hat{\sigma}_{\varepsilon}^{2}

Method	RMSE	MAE
VRMLE	0.5751874	0.4044629
LRCVRMLE	0.7838234	0.6262440
Exact REML	0.5983507	0.4196303
MLE	0.5983492	0.4196293
INLA	0.5982228	0.4195322

Table 2: Predictive performance comparison.

Figure 3: Spatial variation of observed and predicted EPCAM gene count.

Model	Average RMSE (SD)	Average MAE (SD)	Average MSE (SD)
Full Model	0.454 (0.0248)	0.342 (0.0091)	0.207 (0.023)
Reduced Model	0.769 (0.0369)	0.645 (0.019)	0.592 (0.0582)

Table 3: Contribution to the predictive performance of random effects.

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Scalable Sparse Autoregressive Variational Inference for Non-Gaussian Spatial Fields

Paul F. V. Wiemann, Matthias Katzfuss

Department of Statistics, The Ohio State University, Columbus OH 43210, USA · Department of Statistics, University of Wisconsin–Madison, Madison WI 53706, USA

Variational Inference

Open paper AskPDF arXiv

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.

Abstract

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