Evidence Information in Bayesian Updating
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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
perturbation-difference 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 e-information
for 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 these
principles of e-information with the axiomatic Shannon-information theory is performed. Shannoninformation
and developments thereof do not satisfy scale invariance and also violate subcompositional
coherence. In general, Shannon-information theory follows the philosophy of amalgamation
when relating information given by an evidence-vector and some sub-vector, while the dimension
reduction for the proposed e-information corresponds to orthogonal projections in the simplex. The
result of this preliminary study is a set of properties of e-information that may constitute the basis
of an axiomatic theory. A synthetic example is used to motivate the ideas and the subsequent
discussion
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