The distribution of calibrated likelihood functions on the probability-likelihood simplex
Paul-Gauthier Noé
Laboratoire d'informatique et systèmes
https://www.ins2i.cnrs.fr/fr/cnrsinfo/paul-gauthier-noe-et-lapprentissage-statistique
Date(s) : 06/10/2025 iCal
14h00 - 16h00
While calibration of probabilistic predictions has been widely studied
in the fields of weather forecasting, biometrics, statistics, and
machine learning; this presentation will rather address calibration of
likelihood functions. This has been discussed, 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 (log Bayes
factor), We will present the idempotence property and how it leads to a
constraint on their distribution. Especially, if calibrated LLRs are
normally distributed under one hypothesis, they are also normally
distributed under the other hypothesis, with an opposite mean, and a
shared variance equal to twice the mean. Although these results have
been known for decades, they have been limited to the binary case.
In this presentation, we will extend these results to cases with more
than two hypotheses by using the Aitchison geometry of the simplex. The
latter allows us to recover, in a vector form, the additive form of the
Bayes’ rule; extending therefore the concepts of LLR and
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 the multiple
hypotheses and multiclass counterpart of the LLR: the
isometric-log-ratio transformed likelihood function. This work is mainly
conceptual, but if time permits, we will see one application of these
results 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.
Emplacement
I2M Saint-Charles - Salle de séminaire
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