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Neural Likelihood Surfaces for Spatial Processes with Computationally Intensive or Intractable Likelihoods
Methodology · Submitted 8 May 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.99 | 0.05 | 0.02 | speculative 0.49 |
| s1 | In this work, we propose using convolutional neural networks to learn the likelihood function of a spatial process. | contribution 0.80 | 0.85 | 0.03 | asserted 0.94 |
| s2 | 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. | method 0.98 | 0.90 | 0.03 | asserted 0.99 |
| s3 | Once trained on the classification task, our neural network is calibrated using Platt scaling which improves the accuracy of the neural likelihood surfaces. | method 0.99 | 0.94 | 0.03 | asserted 1.00 |
| s4 | 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. | method 0.58 | 0.95 | 0.03 | asserted 0.99 |
| s5 | 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. | interpretation 0.76 | 0.89 | 0.97 | asserted 1.00 |
| s6 | The method is applicable to any spatial process on a grid from which fast simulations are available. | limitation 0.54 | 0.29 | 0.03 | asserted 1.00 |
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The paper
Original abstract
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.
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How the literature uses it
4 arXiv papers cite this in 4 sentences
Papers citing it · 3 sentences without a confident label
Cites it as background 1 paper
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Due to the flexibility and computational scalability of neural networks, recent years have seen a dramatic uptake in their application in the study of spatial extreme values, with their uses ranging from extreme quantile regression [ref] , extremal dependence characterisation [ref] , extreme event simulation [ref] and parameter estimation [ref] .
Neural Bayes estimators for censored inference with peaks-over-threshold models · Introduction · 0.80
What it cites · 5 sentences without a confident label
Cites as background 2 papers
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Beyond improving traditional approaches of SBI, machine learning has ushered in a new approach in which a surrogate for the likelihood or likelihood ratio is learned using training data generated from the simulator [ref] .
Likelihood-Free Frequentist Inference: Bridging Classical Statistics and Machine Learning for Reliable Simulator-Based Inference · Simulation-Based Inference · 0.89
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Another important topic for future work will be to investigate the robustness of our method to input data whose distribution deviates slightly from the stochastic model used to train the network [ref] .
A Systematic Review of Robustness in Deep Learning for Computer Vision: Mind the gap? · Discussion and Conclusions · 0.81
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