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Lower bounds on Bayes factors for multinomial distributions, with application to chi-squared tests of fit. (English) Zbl 0712.62027
Lower bounds on Bayes factors in favour of the null hypothesis in multinomial tests of point null hypotheses are developed. These are then applied to derive lower bounds on Bayes factors in both exact and asymptotic chi-squared testing situations. To develop these bounds, in section 1 a class G of densities g is considered and the choice of this G is made under two situations, such as \(G_{cu}\) (g which are conjugate) and \(G_{us}\) (where g are unimodal symmetric). In section 2, bounds for conjugate priors in multinomial setting are taken up.
Theorem 1 gives the lower bounds \(B_{cu}(n)\) on the Bayes factor over \(G_{cu}\), where \(N=n_ 1+...+n_ t\) and \(n=(n_ 1,...,n_ t)'.\)
Theorem 2 gives the asymptotic behaviour of \(B_{cu}(n)\) and is proved in detail in four steps with the help of four lemmas which are also proved.
Theorem 3 gives the lower bound on the Bayes factor over all conditional densities g in \(G_{tus}\) (where tus stands for transformed unimodal densities).
Theorem 4 proves \(\limsup_{n\to \infty}| B_{tus}(n)-B_{us}(S_ N)| =0.\)
There are 5 tables in the paper. Table 1 gives the asymptotic lower bounds \(B_{cu}\) and \(B_{us}\) for P-values, \(P=.001\),.01,.05 and.10. Tables 2 and 3 tabulate the exact bounds \(B_{cu}\) and \(B_{tus}\) for various values of n and N. Section 5 discusses the chi-square tests of fit and includes tables 4 and 5.
The final section 6 gives a general discussion stating that although the lower bounds \(B_{cu}\) and \(B_{tus}\) seem much more useful than P- values, yet they are just lower bounds. A similar discussion on the specification of g follows.
Reviewer: G.S.Lingappaiah

MSC:
62F15 Bayesian inference
62A01 Foundations and philosophical topics in statistics
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