Inria Montpellier, St-Priest Campus, Building 2, Room 167
Paul-Gauthier Noé LIS – CNRS/ Aix Marseille Université
While calibration of probabilistic predictions has been widely studied, we will rather discuss calibration of likelihood functions. This has been studied, especially in biometrics, in cases with only two exhaustive and mutually exclusive hypotheses (or classes): where likelihood functions can be written as log-likelihood-ratios (LLRs). After defining calibration for LLRs and its connection with the concept of weight-of-evidence, I will present the idempotence property and its associated constraint on the distribution of the LLRs. Although these results have been known for decades, they have been limited to the binary case. In this talk, we will see how the Aitchison geometry of the simplex allows us to extend these results to cases with more than two hypotheses. To be more precise, it recovers, in a vector form, the additive form of the Bayes’ rule; extending therefore the LLR and the weight-of-evidence to any number of hypotheses. Especially, we will extend the definition of calibration, the idempotence, and the constraint on the distribution of likelihood functions to this multiple hypotheses and multiclass counterpart of the LLR: the isometric-log-ratio transformed likelihood function. Even if this work is mainly conceptual, we will discuss one application to machine learning by presenting a non-linear discriminant analysis where the discriminant components form a calibrated likelihood function over the classes, improving therefore the interpretability and the reliability of the method.