In part I of this two-part work, certain minimization problems based on a parametric family of relative entropies (denoted $mathscr{I}_{alpha}$) were studied. Such minimizers were called forward $mathscr{I}_{alpha}$-projections. Here, a complementary
class of minimization problems leading to the so-called reverse $mathscr{I}_{alpha}$-projections are studied. Reverse $mathscr{I}_{alpha}$-projections, particularly on log-convex or power-law families, are of interest in robust estimation problems ($alpha >1$) and in constrained compression settings ($alpha <1$). Orthogonality of the power-law family with an associated linear family is first established and is then exploited to turn a reverse $mathscr{I}_{alpha}$-projection into a forward $mathscr{I}_{alpha}$-projection. The transformed problem is a simpler quasiconvex minimization subject to linear constraints.
We study minimization of a parametric family of relative entropies, termed relative $alpha$-entropies (denoted $mathscr{I}_{alpha}(P,Q)$). These arise as redundancies under mismatched compression when cumulants of compressed lengths are considered in
stead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative $alpha$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimization of $mathscr{I}_{alpha}(P,Q)$ over the first argument on a set of probability distributions that constitutes a linear family is studied. Such a minimization generalizes the maximum R{e}nyi or Tsallis entropy principle. The minimizing probability distribution (termed $mathscr{I}_{alpha}$-projection) for a linear family is shown to have a power-law.
Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative $alpha$-entropies (denoted $mathscr{I}_{alpha}$), arise as redundancies under mismatched comp
ression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative $alpha$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimizers of these relative $alpha$-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum R{e}nyi or Tsallis entropy principle. The minimizing probability distribution (termed forward $mathscr{I}_{alpha}$-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse $mathscr{I}_{alpha}$-projection is studied.
In-plane temperature dependent dielectric behavior of BiFeO3 (BFO) as-grown thin films show diffuse but prominent phase transitions near 450 (+/-10) K and 550 K with dielectric loss temperature dependences that suggest skin layer effects. The 450 K a
nomalies are near the transition first reported by Polomska et al. [Phys. Stat. Sol. 23, 567 (1974)]. The 550 K anomalies coincide with the surface phase transition recently reported [Xavi et al. PRL 106, 236101 (2011)]. In addition, anomalies are found at low temperatures: After several experimental cycles the dielectric loss shows a clear relaxor-like phase transition near what was previously suggested to be a spin reorientation transition (SRT) temperature (~ 201 K) for frequencies 1 kHz < f < 1MHz which follow a nonlinear Vogel-Fulcher (V-F) relation; an additional sharp anomaly is observed near ~180 K at frequencies below 1 kHz. As emphasized recently by Cowley et al. [Adv. Phys. 60, 229 (2011)], skin effects are expected for all relaxor ferroelectrics. Using the interdigital electrodes, experimental data and a theoretical model for in-plane longitudinal and transverse direct magnetoelectric (ME) coefficient are presented.
