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Bayes linear spaces. (English) Zbl 1208.62003
Summary: Linear spaces consisting of $$\sigma$$-finite probability measures and infinite measures (improper priors and likelihood functions) are defined. The commutative group operation, called perturbation, is the updating given by the Bayes theorem; the inverse operation is the Radon-Nikodym derivative. Bayes spaces of measures are sets of classes of proportional measures. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. For example, exponential families appear as affine subspaces with their sufficient statistics as a basis. Bayesian statistics, in particular some well-known properties of conjugated priors and likelihood functions, are revisited and slightly extended.

MSC:
 62A01 Foundations and philosophical topics in statistics 60A10 Probabilistic measure theory 62E10 Characterization and structure theory of statistical distributions
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