Abstract
Regularity conditions for an improper prior function to be regarded as a virtually proper prior density are proposed, and their implications are discussed. The two regularity conditions require that a prior function is defined as a limit of a sequence of proper prior densities and also that the induced posterior density is derived as a smooth limit of the sequence of corresponding posterior densities. This approach is compared with the assumption of a degenerated prior density at an unknown point, which is familiar in the empirical Bayes method. The comparison study extends also to the assumption of an improper prior function discussed separately from any proper prior density. Properties and examples are presented to claim potential usefulness of the proposed notion.
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References
Aitchison, J. (1975). Goodness of prediction fit. Biometrika, 62, 547–554.
Aitkin, M. (1991). Posterior Bayes factors (with discussions). Journal of the Royal Statistical Society, Series B, 53, 111–142.
Aitkin, M. (2009). Statisitical inference: an intergrated Bayesian/likelihood approach. Boca Raton: CRC Press.
Berger, J.O., Bernardo, J.M., Sun, D. (2009). The formal definition of reference priors. The Annals of Statistics, 37, 905–938.
Bolstad V.W. (2007). Introduction to Bayesian statistics. Wiley, New York.
Chacon, J.E., Montanero, J., Nogales, A.G., Perez, P. (2007). On the use of Bayes factor in the frequentist testing of a precise hypothesis. Communications in Statististics -Theory and Methods, 36, 2251–2261.
Corcuera, J.M., Giummole, F. (1999). A generalized Bayes rule for prediction. Scandinavian Journal of Statistics, 26, 265–279.
Ibrahim, J.G., Chen, M.-H. (2000). Power prior distributions for regression models. Statistical Science, 15, 46–60.
Jaynes, E. T. (2003). Probability theory: the logic of science. New York: Cambridge University Press.
Jeffreys, H. (1961). Theory of probability (third edition). Oxford: Oxford University Press.
Jorgensen, B. (1997). The theory of dispersion models. London: Chapman & Hall.
Kass, R.E., Raftery, A.E. (1995). Bayes factors. The Journal of American Statistical Association, 90, 773–795.
Lindley, D.V. (1957). A statistical paradox. Biometrika, 44, 187–192.
McCullagh, P., Han, H. (2011). On Bayes’s theorem for improper mixtures. The Annals of Statistics, 39, 2007–2020.
Plummer, M. (2008). Penalized loss functions for Bayesian model comparison. Biostatisitcs, 9, 523–539.
Speckman, B.P., Sun, D. (2003). Fully Bayesian spline smoothing and intrinsic autoregressive priors. Biometrika, 90, 289–302.
Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A. (2002). Bayesian measures of model complexity and fit (with discussions). Journal of the Royal Statistical Society, Series B, 64, 583–639.
Stone, M. (1977). An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. Journal of the Royal Statistical Society, Series B, 39, 44–47.
Tibshirani, T. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58, 267–288.
Wahba, G. (1985). A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem. The Annals of Statistics, 13, 1378–1402.
Yanagimoto, T. (1991). Estimating a model through the conditional MLE. Annals of the Institute of Statistical Mathematics, 43, 735–746.
Yanagimoto, T., Kashiwagi, N. (1990). Empirical Bayes methods for smoothing data and simultaneous estimation of many parameters. Environmental Health Perspectives, 87, 109–114.
Yanagimoto, T., Ohnishi, T. (2005). Standardized posterior mode for the flexible use of a conjugate prior. Journal of Statistical Planning and Inference, 131, 253–269.
Yanagimoto, T., Ohnishi, T. (2009a). Bayesian prediction of a density function in terms of \(e\)-mixture. Journal of Statistical Planning and Inference, 139, 3064–3075.
Yanagimoto, T., Ohnishi, T. (2009b). Predictive credible region for Bayesian diagnosis of a hypothesis with applications. Journal of the Japan Statistical Society, 39, 111–131.
Yanagimoto, T., Ohnishi, T. (2011). Saddlepoint condition on a predictor to reconfirm the need for the assumption of a prior distribution. Journal of Statistical Planning and Inference, 41, 1990–2000.
Yanagimoto, T., Yanagimoto, M. (1987). The use of the marginal likelihood for a diagnostic test for the goodness of fit of the simple regression model. Technometrics, 29, 95–101.
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The authors are grateful to the two reviewers for their keen comments on the original version, which elucidated points to be clarified.
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Yanagimoto, T., Ohnishi, T. Permissible boundary prior function as a virtually proper prior density. Ann Inst Stat Math 66, 789–809 (2014). https://doi.org/10.1007/s10463-013-0421-1
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DOI: https://doi.org/10.1007/s10463-013-0421-1