# On the Brittleness of Maximum Likelihood Estimation for Gaussian Process Hyperparameter Optimization

arXiv: https://arxiv.org/abs/2608.13793 · HTML: https://mlsanity.com/paper/2608.13793/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 60 references that are in the map.
- First posted on arXiv 13 Aug 2026.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [TabPFN: A Transformer That Solves Small Tabular Classification Problems in a Second](https://mlsanity.com/paper/2207.01848/) (2022)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [A Survey of Transformers](https://mlsanity.com/paper/2106.04554/) (2021)
- [Transformers Can Do Bayesian Inference](https://mlsanity.com/paper/2112.10510/) (2021)
- [TabPFN-2.5: Advancing the State of the Art in Tabular Foundation Models](https://mlsanity.com/paper/2511.08667/) (2025)
- [Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC](https://mlsanity.com/paper/1507.04544/) (2015)
- [Practical Multi-fidelity Bayesian Optimization for Hyperparameter Tuning](https://mlsanity.com/paper/1903.04703/) (2019)
- [PFNs4BO: In-Context Learning for Bayesian Optimization](https://mlsanity.com/paper/2305.17535/) (2023)
- [Dirichlet-based Gaussian Processes for Large-scale Calibrated Classification](https://mlsanity.com/paper/1805.10915/) (2018)
- [Deep Gaussian Processes for Multi-fidelity Modeling](https://mlsanity.com/paper/1903.07320/) (2019)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Latent Map Gaussian Processes for Mixed Variable Metamodeling](https://mlsanity.com/paper/2102.03935/) (2021)
- [Multi-Fidelity Cost-Aware Bayesian Optimization](https://mlsanity.com/paper/2211.02732/) (2022)
- [GP+: A Python Library for Kernel-based learning via Gaussian Processes](https://mlsanity.com/paper/2312.07694/) (2023)
- [Data Fusion with Latent Map Gaussian Processes](https://mlsanity.com/paper/2112.02206/) (2021)
- [Unveiling Processing--Property Relationships in Laser Powder Bed Fusion: The Synergy of Machine Learning and High-throughput Experiments](https://mlsanity.com/paper/2409.00248/) (2024)

## Abstract (arXiv)

Machine learning (ML) has become an indispensable part of modern engineering design workflows. A crucial step in training an ML model is the selection of the loss function which can be systematically formulated via various techniques such as maximum likelihood estimation (MLE) and cross-validation . While MLE is one of the most popular, effective, and intuitive mechanisms for training ML models, it is brittle: if the assumptions underpinning it are not met, the trained ML model may generalize poorly. This brittleness affects even Gaussian processes (GPs) which are widely used in engineering design and are often (incorrectly) presumed to be very robust to overfitting. In this paper, we fundamentally evaluate the brittleness of MLE in the context of training GPs for probabilistic regression or classification tasks. We compare theoretically grounded metrics against MLE and propose practical solutions. Our extensive studies demonstrate the effectiveness of our solutions in downstream design tasks such as Bayesian optimization and provide a blueprint for practitioners to build accurate and robust GPs that can even outperform tabular foundation models in terms of prediction accuracy, uncertainty quantification, and inference cost. Our contributions are publicly available via GitHub at https://github.com/Bostanabad-Research-Group/GP-vs-TabPFN-vs-GPyTorch.