  Pyrochlore dielectric materials such as
(Bi$_{1.5}$Zn$_{0.5}$)(Nb$_{1.5}$Zn$_{0.5}$)O$_7$ (BZN) have generated interest
because they combine high dielectric constants with small dielectric loss
tangents and yet are cubic at all temperatures. The recent low-temperature
preparation and structural characterization of Bi$_2$Ti$_2$O$_7$, which remains
cubic down to 2 K, has provided a good model system for understanding the
properties of Bi-based pyrochlores. In this contribution, the electronic
structure of cubic Bi$_2$Ti$_2$O$_7$ is visualized and compared with the
electronic structure of the Aurivillius phase ferroelectric SrBi$_2$Ta$_2$O$_9$
(SBT), which displays a ferroelectric distortion below 608 K associated with
the tendency of lone pair active Bi$^{3+}$ to move off-center. Such coherent
off-centering distortions are frustrated on the pyrochlore lattice, and this
prevents a ferroelectric-paraelectric phase transition in Bi$_2$Ti$_2$O$_7$.
Instead, Bi$^{3+}$ ions in Bi$_2$Ti$_2$O$_7$ are obliged to off-center in an
\textit{incoherent} manner, that is compatible with the cubic structure being
retained. Frustrated lone pair behavior in the defect pyrochlore
Pb$_2$Sn$_2$O$_6$ is also described. Parallels between the well-studied
frustration of certain types of \textit{magnetism} in pyrochlore compounds
(spin-ice) and the striking paucity of ferroelectric pyrochlores, arising from
the corner-connected tetrahedral topology of the pyrochlore lattice are pointed
out.
