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On conditional independence and the relationship between sufficiency and invariance under the Bayesian point of view. (English) Zbl 0964.62003
The paper presents, from a Bayesian point of view, several results about conditional independence and about its role in analyzing the relations between concepts of sufficiency and concepts of invariance. Some of the given results are, in particular, related with the Bayesian translation of the “Stein theorem” [see {\it W.J. Hall}, {\it R.A. Wijsman} and {\it J.K. Ghosh}, Ann. Math. Stat. 36, 575-614 (1965; Zbl 0227.62007)], which gives sufficient conditions under which ${\cal A}_S\cap {\cal A}_I$ is sufficient for ${\cal A}_I$, where ${\cal A}_S$ is a sufficient $\sigma$-field and ${\cal A}_I$ is the $\sigma$-field of all events, invariant under the action of a given group $G$. The paper starts with a brief review about concepts of sufficiency and invariance (both in the classic and Bayesian statistics settings) and ends with some further discussion about the Stein theorem. Some examples of the theory are provided.

##### MSC:
 62A01 Foundations and philosophical topics in statistics 62B05 Sufficient statistics and fields
Full Text:
##### References:
 [1] Berk, R. H.: A note on sufficiency and invariance.. Ann. math. Statist. 43, No. 2, 647-650 (1972) · Zbl 0254.62002 [2] Dawid, A. P.: Some misleading arguments involving conditional independence.. The journal of the royal statistical society, series B 41, 249-252 (1979) [3] Dawid, A. P.: Conditional independence in statistical theory.. The journal of the royal statistical society, series B 41, 1-31 (1979) · Zbl 0408.62004 [4] Florens, J.P., Mouchart, M., Rolin, J.M., 1990. Elements of Bayesian Statistics. Marcel Dekker Inc., New York, 1990. · Zbl 0697.62002 [5] Hall, W. J.; Wijsman, R. A.; Ghosh, J. R.: The relationship between sufficiency and invariance with applications in sequential analysis.. Ann. math. Statist. 36, 575-614 (1965) · Zbl 0227.62007 [6] Landers, D.; Rogge, L.: On sufficiency and invariance.. Ann. statist. 1, No. 3, 543-544 (1973) · Zbl 0258.62005 [7] Nogales, A.G., Oyola, J.A., 1996. Some remarks on sufficiency, invariance and conditional independence. Ann. Statist. 24 (2). · Zbl 0859.62009