Information concept in filtered experiments. (English) Zbl 1057.94007

Dzhaparidze, K.; Spreij, P.; Valkeila, E.

Teor. Jmovirn. Mat. Stat. 67, 38-56 (2002) and Theory Probab. Math. Stat. 67, 43-62 (2003).

The authors investigate the Kullbach-Leibler information (KLI) between a posterior and prior distributions on an abstract parameter set in connection with a concept of randomised filter experiments. In the beginning of the paper they treat the randomised statistical experiments in a static set up. Then they switch to a dynamic setting and analyse the dynamics of KLI in experiments equipped with filtration. For a proper analysis of the dynamic information, the notion of arithmetic and geometric mean measures are used. The latter are naturally linked with a generalized Hellinger process with respect to an arbitrary, not necessarily finite, family of probability measures. Some extremal properties of arithmetic and geometric measures are presented. To illustrate the dynamical behaviour of the KLI the authors give its representation in terms of the likelihood ratio between the arithmetic and geometric measures and the Hellinger process. Therefore a considerable part of the present paper is devoted to the Hellinger process. The martingale and sub-martingale technique is also intensively used.

Reviewer: N. M. Zinchenko (Kyïv)

MSC:

94A17	Measures of information, entropy
62B10	Statistical aspects of information-theoretic topics

Keywords:

Kullbach-Leibler information; Hellinger process; sub-martingale