Geodesic
$\phi $PHI-divergences, sufficiency, Bayes sufficiency, and deficiency
Kybernetika, Tome 48 (2012) no. 4, pp. 690-713

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

The paper studies the relations between $\phi$-divergences and fundamental concepts of decision theory such as sufficiency, Bayes sufficiency, and LeCam's deficiency. A new and considerably simplified approach is given to the spectral representation of $\phi $-divergences already established in Österreicher and Feldman [28] under restrictive conditions and in Liese and Vajda [22], [23] in the general form. The simplification is achieved by a new integral representation of convex functions in terms of elementary convex functions which are strictly convex at one point only. Bayes sufficiency is characterized with the help of a binary model that consists of the joint distribution and the product of the marginal distributions of the observation and the parameter, respectively. LeCam's deficiency is expressed in terms of $\phi $-divergences where $\phi $ belongs to a class of convex functions whose curvature measures are finite and satisfy a normalization condition.
Classification : 62B05, 62B10, 62B15, 62G10
Keywords: divergences; sufficiency; Bayes sufficiency; deficiency 
@article{KYB_2012__48_4_a4,
     author = {Liese, Friedrich},
     title = {$\phi ${PHI-divergences,} sufficiency, {Bayes} sufficiency, and deficiency},
     journal = {Kybernetika},
     pages = {690--713},
     publisher = {mathdoc},
     volume = {48},
     number = {4},
     year = {2012},
     mrnumber = {3013395},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2012__48_4_a4/}
}
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Liese, Friedrich. $\phi $PHI-divergences, sufficiency, Bayes sufficiency, and deficiency. Kybernetika, Tome 48 (2012) no. 4, pp. 690-713. http://geodesic.mathdoc.fr/item/KYB_2012__48_4_a4/