# PriorVAE: Encoding spatial priors with VAEs for small-area estimation

arXiv: https://arxiv.org/abs/2110.10422 · HTML: https://mlsanity.com/paper/2110.10422/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 45 references that are in the map.
- First posted on arXiv 20 Oct 2021.

## Builds on (most cited first, 11 of 11)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Constrained Graph Variational Autoencoders for Molecule Design](https://mlsanity.com/paper/1805.09076/) (2018)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [GP-VAE: Deep Probabilistic Time Series Imputation](https://mlsanity.com/paper/1907.04155/) (2019)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [Yes, but Did It Work?: Evaluating Variational Inference](https://mlsanity.com/paper/1802.02538/) (2018)
- [An intuitive Bayesian spatial model for disease mapping that accounts for scaling](https://mlsanity.com/paper/1601.01180/) (2016)
- [A Spatially Discrete Approximation to Log-Gaussian Cox Processes for Modelling Aggregated Disease Count Data](https://mlsanity.com/paper/1901.09551/) (2019)

## Built on by (most cited first, 11 of 11)

- [PriorCVAE: scalable MCMC parameter inference with Bayesian deep generative modelling](https://mlsanity.com/paper/2304.04307/) (2023)
- [SMRS: advocating a unified reporting standard for surrogate models in the artificial intelligence era](https://mlsanity.com/paper/2502.06753/) (2025)
- [Cox-Hawkes: doubly stochastic spatiotemporal Poisson processes](https://mlsanity.com/paper/2210.11844/) (2022)
- [Predictive posterior sampling from non-stationnary Gaussian process priors via Diffusion models with application to climate data](https://mlsanity.com/paper/2505.24556/) (2025)
- [HawkesNest: A Multi-Axis Synthetic Benchmark for Spatiotemporal Pattern Complexity](https://mlsanity.com/paper/2606.16863/) (2026)
- [Deep learning and MCMC with aggVAE for shifting administrative boundaries: mapping malaria prevalence in Kenya](https://mlsanity.com/paper/2305.19779/) (2023)
- [Federated Learning for Non-factorizable Models using Deep Generative Prior Approximations](https://mlsanity.com/paper/2405.16055/) (2024)
- [Distance-Preserving Representations for Genomic Spatial Reconstruction](https://mlsanity.com/paper/2408.00911/) (2024)
- [Variational Autoencoded Multivariate Spatial Fay-Herriot Models](https://mlsanity.com/paper/2503.14710/) (2025)
- [DeepRV: Accelerating Spatiotemporal Inference with Pre-trained Neural Priors](https://mlsanity.com/paper/2503.21473/) (2025)
- [DiffKnock: Diffusion-based Knockoff Statistics for Neural Networks Inference](https://mlsanity.com/paper/2510.01418/) (2025)

## Abstract (arXiv)

Gaussian processes (GPs), implemented through multivariate Gaussian distributions for a finite collection of data, are the most popular approach in small-area spatial statistical modelling. In this context they are used to encode correlation structures over space and can generalise well in interpolation tasks. Despite their flexibility, off-the-shelf GPs present serious computational challenges which limit their scalability and practical usefulness in applied settings. Here, we propose a novel, deep generative modelling approach to tackle this challenge, termed PriorVAE: for a particular spatial setting, we approximate a class of GP priors through prior sampling and subsequent fitting of a variational autoencoder (VAE). Given a trained VAE, the resultant decoder allows spatial inference to become incredibly efficient due to the low dimensional, independently distributed latent Gaussian space representation of the VAE. Once trained, inference using the VAE decoder replaces the GP within a Bayesian sampling framework. This approach provides tractable and easy-to-implement means of approximately encoding spatial priors and facilitates efficient statistical inference. We demonstrate the utility of our VAE two stage approach on Bayesian, small-area estimation tasks.