# Post-hoc Calibration of Neural Networks by g-Layers

arXiv: https://arxiv.org/abs/2006.12807 · HTML: https://mlsanity.com/paper/2006.12807/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2020.

## Built on by (most cited first, 5 of 5)

- [How to Fix a Broken Confidence Estimator: Evaluating Post-hoc Methods for Selective Classification with Deep Neural Networks](https://mlsanity.com/paper/2305.15508/) (2023)
- [On Calibration of Object Detectors: Pitfalls, Evaluation and Baselines](https://mlsanity.com/paper/2405.20459/) (2024)
- [The Robust Semantic Segmentation UNCV2023 Challenge Results](https://mlsanity.com/paper/2309.15478/) (2023)
- [Instance-Wise Monotonic Calibration by Constrained Transformation](https://mlsanity.com/paper/2507.06516/) (2025)
- [Adaptive Calibration for Fair and Performant Facial Recognition](https://mlsanity.com/paper/2606.04469/) (2026)

## Abstract (arXiv)

Calibration of neural networks is a critical aspect to consider when incorporating machine learning models in real-world decision-making systems where the confidence of decisions are equally important as the decisions themselves. In recent years, there is a surge of research on neural network calibration and the majority of the works can be categorized into post-hoc calibration methods, defined as methods that learn an additional function to calibrate an already trained base network. In this work, we intend to understand the post-hoc calibration methods from a theoretical point of view. Especially, it is known that minimizing Negative Log-Likelihood (NLL) will lead to a calibrated network on the training set if the global optimum is attained (Bishop, 1994). Nevertheless, it is not clear learning an additional function in a post-hoc manner would lead to calibration in the theoretical sense. To this end, we prove that even though the base network (f) does not lead to the global optimum of NLL, by adding additional layers (g) and minimizing NLL by optimizing the parameters of g one can obtain a calibrated network g circ f. This not only provides a less stringent condition to obtain a calibrated network but also provides a theoretical justification of post-hoc calibration methods. Our experiments on various image classification benchmarks confirm the theory.