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Advances in Logic Based Intelligent Systems: Selected Papers by K. Nakamatsu, M. Abe (Editors)

By K. Nakamatsu, M. Abe (Editors)

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We are interested in precise hypotheses, with dim(Θ0 ) < dim(Θ) . f (θ) is the posterior probability density function. The computation of the evidence measure used on the FBST is performed in two steps: The optimization step consists of finding f ∗ , the maximum (supremum) of the posterior under the null hypothesis. , Ev(H) = Pr(θ ∈ T | x) = f (θ)dθ , where T ∗ T = {θ ∈ Θ : f (θ) > f } and f ∗ = supH f (θ) Ev(H) is the evidence against H, and Ev(H) = 1 − Ev(H) is the evidence supporting (or in favour of) H.

Anderson (1935). The Irises of the Gaspé Peninsula. Bulletin of the American Iris Society, 59, 2-5. Peel (2000). Finite Mixture Models. NY: Wiley. Robert (1996). Mixture of Distributions: Inference and Estimation. In [3]. Zacks (2003). Comparing Parameters of Two Bivariate Normal Distributions Using the Invariant FBST. Brazilian Journal of Probability and Statistics, 17, 147-168. Stern (2003). Bayesian Evidence Test for Precise Hypotheses. Journal of Statistical Planning and Inference, 117,185–198.

Atkinson [30], although in a very different theoretical framework, was the first to analyse this problem using a mixture formulation, f (x | θ) = m k=1 wk fk (x, ψk ) . The theory for mixture models presented here can be adapted to analyse the problem of discriminating between separate hypotheses. This is the subject of the authors’ forthcoming articles with Carlos Alberto de Bragança Pereira and Basílio de Bragança Pereira. The authors are grateful for support of Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Apoio à Pesquisa do Estado de São Paulo (FAPESP).

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