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Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Egozcue, Juan JosÃ©
PawlowskyGlahn, Vera 

2011 May 12  
Bayes theorem (discrete case) is taken as a paradigm of information acquisition. As mentioned by Aitchison, Bayes formula can be identified with perturbation of a prior probability vector and a discrete likelihood function, both vectors being compositional. Considering prior, posterior and likelihood as elements of the simplex, a natural choice of distance between them is the Aitchison 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 the perturbationdifference to think of opposite information. The Aitchison norm provides a size of compositional vectors, and is thus a natural scalar measure of the information conveyed by the likelihood or captured by a prior or a posterior. It is called evidence information, or einformation for short. In order to support such einformation theory some principles of einformation are discussed. They essentially coincide with those of compositional data analysis. Also, a comparison of these principles of einformation with the axiomatic Shannoninformation theory is performed. Shannoninformation and developments thereof do not satisfy scale invariance and also violate subcompositional coherence. In general, Shannoninformation theory follows the philosophy of amalgamation when relating information given by an evidencevector and some subvector, while the dimension reduction for the proposed einformation corresponds to orthogonal projections in the simplex. The result of this preliminary study is a set of properties of einformation that may constitute the basis of an axiomatic theory. A synthetic example is used to motivate the ideas and the subsequent discussion  
application/pdf  
http://hdl.handle.net/10256/13632  
eng  
Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
CoDaWork 2011. The 4th International Workshop on Compositional Data Analysis  
Tots els drets reservats  
EstadÃstica matemÃ tica  Congressos
Mathematical statistics  Congresses AnÃ lisi multivariable  Congressos Multivariate analysis  Congresses EstadÃstica bayesiana  Congressos Bayesian statistical decision theory  Congresses 

Evidence Information in Bayesian Updating  
info:eurepo/semantics/conferenceObject  
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