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 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.
Citationvan den Boogaart, K.; Egozcue, J. J.; Pawlowsky-Glahn, V. Bayes linear spaces. "SORT: statistics and operations research transactions", Desembre 2010, vol. 34, núm. 2, p. 201-222.
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