# Neural Likelihood Surfaces for Spatial Processes with Computationally Intensive or Intractable Likelihoods

arXiv: https://arxiv.org/abs/2305.04634 · HTML: https://mlsanity.com/paper/2305.04634/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 8 May 2023.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [A Systematic Review of Robustness in Deep Learning for Computer Vision: Mind the gap?](https://mlsanity.com/paper/2112.00639/) (2021)
- [A Case Study Competition Among Methods for Analyzing Large Spatial Data](https://mlsanity.com/paper/1710.05013/) (2017)
- [Likelihood-Free Parameter Estimation with Neural Bayes Estimators](https://mlsanity.com/paper/2208.12942/) (2022)
- [Neural Networks for Parameter Estimation in Intractable Models](https://mlsanity.com/paper/2107.14346/) (2021)
- [Fast covariance parameter estimation of spatial Gaussian process models using neural networks](https://mlsanity.com/paper/2012.15339/) (2020)
- [Confidence Sets and Hypothesis Testing in a Likelihood-Free Inference Setting](https://mlsanity.com/paper/2002.10399/) (2020)
- [Simulation-based inference methods for particle physics](https://mlsanity.com/paper/2010.06439/) (2020)
- [Locally stationary spatio-temporal interpolation of Argo profiling float data](https://mlsanity.com/paper/1711.00460/) (2017)

## Built on by (most cited first, 10 of 10)

- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks](https://mlsanity.com/paper/2310.02600/) (2023)
- [Neural Bayes estimators for censored inference with peaks-over-threshold models](https://mlsanity.com/paper/2306.15642/) (2023)
- [Theoretical guarantees for neural estimators in parametric statistics](https://mlsanity.com/paper/2506.18508/) (2025)
- [Modeling Spatial Extremes using Non-Gaussian Spatial Autoregressive Models via Convolutional Neural Networks](https://mlsanity.com/paper/2505.03034/) (2025)
- [LatticeVision: Image to Image Networks for Modeling Non-Stationary Spatial Data](https://mlsanity.com/paper/2505.09803/) (2025)
- [A Generalized Unified Skew-Normal Process with Neural Bayes Inference](https://mlsanity.com/paper/2411.17400/) (2024)
- [Neural Bayes estimation and selection for complex bivariate extremal dependence models](https://mlsanity.com/paper/2503.23156/) (2025)
- [Keeping Score: Adaptive, Tuning-Free Loss Weighting for Score-Augmented Neural Ratio Estimation](https://mlsanity.com/paper/2605.12118/) (2026)
- [A Convex Approximation Framework for Neural Likelihood-Based Bayesian Inverse Problems](https://mlsanity.com/paper/2607.06252/) (2026)

## Abstract (arXiv)

In spatial statistics, fast and accurate parameter estimation, coupled with a reliable means of uncertainty quantification, can be challenging when fitting a spatial process to real-world data because the likelihood function might be slow to evaluate or wholly intractable. In this work, we propose using convolutional neural networks to learn the likelihood function of a spatial process. Through a specifically designed classification task, our neural network implicitly learns the likelihood function, even in situations where the exact likelihood is not explicitly available. Once trained on the classification task, our neural network is calibrated using Platt scaling which improves the accuracy of the neural likelihood surfaces. To demonstrate our approach, we compare neural likelihood surfaces and the resulting maximum likelihood estimates and approximate confidence regions with the equivalent for exact or approximate likelihood for two different spatial processes: a Gaussian process and a Brown-Resnick process which have computationally intensive and intractable likelihoods, respectively. We conclude that our method provides fast and accurate parameter estimation with a reliable method of uncertainty quantification in situations where standard methods are either undesirably slow or inaccurate. The method is applicable to any spatial process on a grid from which fast simulations are available.