  We study data processing inequalities that are derived from a certain class
of generalized information measures, where a series of convex functions and
multiplicative likelihood ratios are nested alternately. While these
information measures can be viewed as a special case of the most general
Zakai-Ziv generalized information measure, this special nested structure calls
for attention and motivates our study. Specifically, a certain choice of the
convex functions leads to an information measure that extends the notion of the
Bhattacharyya distance (or the Chernoff divergence): While the ordinary
Bhattacharyya distance is based on the (weighted) geometric mean of two
replicas of the channel's conditional distribution, the more general
information measure allows an arbitrary number of such replicas. We apply the
data processing inequality induced by this information measure to a detailed
study of lower bounds of parameter estimation under additive white Gaussian
noise (AWGN) and show that in certain cases, tighter bounds can be obtained by
using more than two replicas. While the resulting lower bound may not compete
favorably with the best bounds available for the ordinary AWGN channel, the
advantage of the new lower bound, relative to the other bounds, becomes
significant in the presence of channel uncertainty, like unknown fading. This
different behavior in the presence of channel uncertainty is explained by the
convexity property of the information measure.
