×

zbMATH — the first resource for mathematics

On the existence of inferences which are consistent with a given model. (English) Zbl 0866.62001
Summary: If \(\{p_\theta\}\) is a \(\sigma\)-additive statistical model and \(\pi\) a finitely additive prior, then any statistic \(T\) is sufficient, with respect to a suitable inference consistent with \(\{p_\theta\}\) and \(\pi\), provided only that \(p_\theta(T=t)=0\) for all \(\theta\) and \(t\). Here, sufficiency is to be intended in the Bayesian sense, and consistency in the sense of D. A. Lane and W. D. Sudderth [ibid. 11, 114-120 (1983; Zbl 0555.62008)]. As a corollary, if \(\{p_\theta\}\) is \(\sigma\)-additive and diffuse, then, whatever the prior \(\pi\), there is an inference which is consistent with \(\{p_\theta\}\) and \(\pi\). Two versions of the main result are also obtained for predictive problems.

MSC:
62A01 Foundations and philosophical topics in statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] BARANCHIK, A. J. 1970. A family of minimax estimators of the mean of a multivariate normal distribution. Ann. Math. Statist. 41 642 645. Z.
[2] BERTI, P., REGAZZINI, E. and RIGO, P. 1991. Coherent statistical inference and Bay es theorem. Ann. Statist. 19 366 381. Z. · Zbl 0742.62003
[3] BERTI, P. and RIGO, P. 1992. Weak disintegrability as a form of preservation of coherence. Journal of the Italian Statistical Society 1 161 181. Z. · Zbl 1446.60003
[4] BERTI, P. and RIGO, P. 1994. Coherent inferences and improper priors. Ann. Statist. 22 1177 1194. Z. · Zbl 0827.62003
[5] BLACKWELL, D. 1955. On a class of probability spaces. Proc. Third Berkeley Sy mp. Math. Statist. Probab. 2 1 6. Univ. California Press, Berkeley. Z. · Zbl 0073.12301
[6] BLACKWELL, D. and RAMAMOORTHI, R. V. 1982. A Bay es but not classically sufficient statistic. Ann. Statist. 10 1025 1026. Z. · Zbl 0485.62004
[7] BROWN, L. D. and PURVES, R. 1973. Measurable selections of extrema. Ann. Statist. 1 902 912. Z. · Zbl 0265.28003
[8] CIFARELLI, D. M. and REGAZZINI, E. 1982. Some considerations about mathematical statistics teaching methodology suggested by the concept of exchangeability. In Exchangeability Z. in Probability and Statistics G. Koch and F. Spizzichino, eds. 185 205. North-Holland, Amsterdam. Z. · Zbl 0491.62006
[9] DUBINS, L. E. 1975. Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Probab. 3 89 99. Z. · Zbl 0302.60002
[10] DUNFORD, N. and SCHWARTZ, J. T. 1958. Linear Operators, Part I: General Theory. Interscience, New York. Z. · Zbl 0084.10402
[11] HEATH, D. and SUDDERTH, W. D. 1978. On finitely additive priors, coherence, and extended admissibility. Ann. Statist. 6 333 345. Z. · Zbl 0385.62005
[12] HEATH, D. and SUDDERTH, W. D. 1989. Coherent inference from improper priors and from finitely additive priors. Ann. Statist. 17 907 919. Z. · Zbl 0687.62003
[13] JAMES, W. and STEIN, C. 1961. Estimation with quadratic loss. Proc. Fourth Berkeley Sy mp. Math. Statist. Probab. 1 361 379. Univ. California Press, Berkeley. Z. · Zbl 1281.62026
[14] LANE, D. A. and SUDDERTH, W. D. 1983. Coherent and continuous inference. Ann. Statist. 11 114 120. · Zbl 0563.62003
[15] LANE, D. A. and SUDDERTH, W. D. 1984. Coherent predictive inference. Sankhy a Ser. A 46 166 185. Z. · Zbl 0574.62002
[16] PARTHASARATHY, K. R. 1967. Probability Measures on Metric Spaces. Academic, New York. Z. · Zbl 0153.19101
[17] PRIKRY, K. and SUDDERTH, W. D. 1982. Singularity with respect to strategic measures. Illinois J. Math. 26 460 465. Z. · Zbl 0472.60005
[18] REGAZZINI, E. 1987. De Finetti’s coherence and statistical inference. Ann. Statist. 15 845 864. Z. · Zbl 0653.62003
[19] STEIN, C. 1955. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Sy mp. Math. Statist. Probab. 1 197 206. Univ. California Press, Berkeley. Z. · Zbl 0073.35602
[20] SUDDERTH, W. D. 1980. Finitely additive priors, coherence and the marginalization paradox. J. Roy. Statist. Soc. Ser. B 42 339 341. Z. JSTOR: · Zbl 0442.62005
[21] WETZEL, N. R. 1993. Coherent inferences for multivariate data models. Ph.D. dissertation, School Statist., Univ. Minnesota, Minneapolis.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.