DUGi

Evidence Information in Bayesian Updating

http://dugi.udg.edu/item/http:@@@@hdl.handle.net@@2072@@273653

Bayes theorem (discrete case) is taken as a paradigm of information acquisition. As mentionedby Aitchison, Bayes formula can be identified with perturbation of a prior probability vectorand a discrete likelihood function, both vectors being compositional. Considering prior, posteriorand likelihood as elements of the simplex, a natural choice of distance between them is theAitchison distance. Other geometrical features can also be considered using the Aitchison geometry.For instance, orthogonality in the simplex allows to think of orthogonal information, or theperturbation-difference to think of opposite information. The Aitchison norm provides a size ofcompositional vectors, and is thus a natural scalar measure of the information conveyed by thelikelihood or captured by a prior or a posterior. It is called evidence information, or e-informationfor short.In order to support such e-information theory some principles of e-information are discussed.They essentially coincide with those of compositional data analysis. Also, a comparison of theseprinciples of e-information with the axiomatic Shannon-information theory is performed. Shannoninformationand developments thereof do not satisfy scale invariance and also violate subcompositionalcoherence. In general, Shannon-information theory follows the philosophy of amalgamationwhen relating information given by an evidence-vector and some sub-vector, while the dimensionreduction for the proposed e-information corresponds to orthogonal projections in the simplex. Theresult of this preliminary study is a set of properties of e-information that may constitute the basisof an axiomatic theory. A synthetic example is used to motivate the ideas and the subsequentdiscussion

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada