  Inorganic clathrates such as Ba$_8$Ga$_{x}$Ge$_{46-x}$ and
Ba$_8$Al$_{x}$Si$_{46-x}$ commonly exhibit very low thermal conductivities. A
quantitative computational description of this important property has proven
difficult, in part due to the large unit cell, the role of disorder, and the
fact that both electronic carriers and phonons contribute to transport. Here,
we conduct a systematic analysis of the temperature and composition dependence
of low-frequency modes associated with guest species in
Ba$_8$Ga$_{x}$Ge$_{46-x}$ and Ba$_8$Al$_{x}$Si$_{46-x}$ ("rattler modes"), as
well as of thermal transport in stoichiometric Ba$_8$Ga$_{16}$Ge$_{30}$. To
this end, we account for phonon-phonon interactions by means of temperature
dependent effective interatomic force constants (TDIFCs), which we find to be
crucial in order to achieve an accurate description of the lattice part of the
thermal conductivity. While the analysis of the thermal conductivity is often
largely focused on the rattler modes, here, it is shown that at room
temperatures modes with $\hbar\omega\gtrsim\,10\,\text{meV}$ account for 50\%\
of lattice heat transport. Finally, the electronic contribution to the thermal
conductivity is computed, which shows the Wiedemann-Franz law to be only
approximately fulfilled. As a result, it is crucial to employ the correct
prefactor when separating electronic and lattice contributions for experimental
data.
