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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.

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