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On the invariance of noninformative priors. (English) Zbl 0906.62024

Summary: H. Jeffreys’ prior [see “Theory of probability.” 3rd ed. (1961; Zbl 0116.34904)], one of the widely used noninformative priors, remains invariant under reparametrization, but does not perform satisfactorily in the presence of nuisance parameters. To overcome this deficiency, recently various noninformative priors have been proposed in the literature.
This article explores the invariance (or lack thereof) of some of these noninformative priors including the reference prior of J. O. Berger and J. M. Bernardo [J. Am. Stat. Assoc. 84, No. 405, 200-207 (1989; Zbl 0682.62018)], the reverse reference prior of J. K. Ghosh [see J. K. Ghosh and R. Mukerjee, in J. M. Bernardo et al. (eds.), Bayesian Statistics 4, 195-200 (1992)] and the probability-matching prior of H. W. Peers [J. R. Stat. Soc., Ser. B 27, 9-16 (1965; Zbl 0144.41403)] and C. M. Stein [Sequential Methods in Statistics, Banach Cent. Publ. 16, 485-514 (1985; Zbl 0662.62030)] under reparametrization. Berger and Bernardo’s \(m\)-group ordered reference prior is shown to remain invariant under a special type of reparametrization. The reverse reference prior of J. K. Ghosh is shown not to remain invariant under reparameterization. However, the probability-matching prior is shown to remain invariant under any reparametrization. Also for spherically symmetric distributions, certain noninformative priors are derived using the principle of group invariance.

MSC:

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

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