Given a (finitely additive) full conditional probability space and a conditional measurable space , a multivalued mapping Γ from X to Y induces a class of full conditional probabilities on . A closed form expression for the lower and upper envelopes ⁎ and ⁎ of such class is provided: the envelopes can be expressed through a generalized Bayesian conditioning rule, relying on two linearly ordered classes of (possibly unbounded) inner and outer measures. For every , ⁎ is a normalized totally monotone capacity which is continuous from above if is a countably additive full conditional probability space and is a σ-algebra. Moreover, the full conditional prevision functional M induced by μ on the set of -continuous conditional gambles is shown to give rise through Γ to the lower and upper full conditional prevision functionals ⁎ and ⁎ on the set of -continuous conditional gambles.
Upper and lower conditional probabilities induced by a multivalued mapping / Petturiti, Davide; Vantaggi, Barbara. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 458:2(2018), pp. 1214-1235. [10.1016/j.jmaa.2017.10.006]
Upper and lower conditional probabilities induced by a multivalued mapping
Vantaggi, Barbara
2018
Abstract
Given a (finitely additive) full conditional probability space and a conditional measurable space , a multivalued mapping Γ from X to Y induces a class of full conditional probabilities on . A closed form expression for the lower and upper envelopes ⁎ and ⁎ of such class is provided: the envelopes can be expressed through a generalized Bayesian conditioning rule, relying on two linearly ordered classes of (possibly unbounded) inner and outer measures. For every , ⁎ is a normalized totally monotone capacity which is continuous from above if is a countably additive full conditional probability space and is a σ-algebra. Moreover, the full conditional prevision functional M induced by μ on the set of -continuous conditional gambles is shown to give rise through Γ to the lower and upper full conditional prevision functionals ⁎ and ⁎ on the set of -continuous conditional gambles.File | Dimensione | Formato | |
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