  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.
