Laplace's rule of succession in information geometry

28/10/2015
Auteurs : Yann Ollivier
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14288
DOI : http://dx.doi.org/10.1007/978-3-319-25040-3_34You do not have permission to access embedded form.

Résumé

When observing data x1, . . . , x t modelled by a probabilistic distribution pθ(x), the maximum likelihood (ML) estimator θML = arg max θ Σti=1 ln pθ(x i ) cannot, in general, safely be used to predict xt + 1. For instance, for a Bernoulli process, if only “tails” have been observed so far, the probability of “heads” is estimated to 0. (Thus for the standard log-loss scoring rule, this results in infinite loss the first time “heads” appears.)

Laplace's rule of succession in information geometry

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When observing data x1, . . . , x t modelled by a probabilistic distribution pθ(x), the maximum likelihood (ML) estimator θML = arg max θ Σti=1 ln pθ(x i ) cannot, in general, safely be used to predict xt + 1. For instance, for a Bernoulli process, if only “tails” have been observed so far, the probability of “heads” is estimated to 0. (Thus for the standard log-loss scoring rule, this results in infinite loss the first time “heads” appears.)

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