Frequentistic approximations to Bayesian prevision of exchangeable random elements |
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Institution: | 1. Dipartimento di Scienze delle Decisioni, Università commerciale L. Bocconi, 20136 Milano, Italy;2. Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, 41125 Modena, Italy;3. Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy;4. CNR-IMATI, Milano, Italy |
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Abstract: | Posterior and predictive distributions for m future trials, given the first n elements of an infinite exchangeable sequence , are considered in a nonparametric Bayesian setting. The former distribution is compared to the unit mass at the empirical distribution of the n past observations, while the latter is compared to the m-fold product . Comparisons are made by means of distinguished probability distances inducing topologies that are equivalent to (or finer than) the topology of weak convergence of probability measures. After stating almost sure convergence to zero of these distances as n goes to infinity, the paper focuses on the analysis of the rate of approach to zero, so providing a quantitative evaluation of the approximation of posterior and predictive distributions through their frequentistic counterparts and , respectively. Characteristic features of the present work, with respect to more common literature on Bayesian consistency, are: first, comparisons are made between entities which depend on the n past observation only; second, the approximations are studied under the actual (exchangeable) law of the 's, and not under hypothetical product laws , as varies among the admissible determinations of a random probability measure. |
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Keywords: | Asymptotics of prior and posterior distributions Exchangeability Inverse probabilities Probability distances Random measures |
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