  We use vibrational dynamical mean-field theory (VDMFT) to study the
vibrational structure of type-I clathrate solids, specifically
X$_8$Ga$_{16}$Ge$_{30}$, where X=Ba,Sr. These materials are cage-like chemical
structures hosting loosely bound guest atoms, resulting in strong
anharmonicity, short phonon lifetimes, and ultra-low thermal conductivities.
Presenting the methodological developments necessary for this first application
to three-dimensional, atomistic materials, we validate our approach through
comparison to molecular dynamics simulations and show that VDMFT is extremely
accurate at a fraction of the cost. Through the use of nonperturbative methods,
we find that anharmonicity is dominated by four-phonon and higher-order
scattering processes, and it causes rattler modes to shift up in frequency by
50% (10 cm$^{-1}$) and to have lifetimes of less than 1 ps; this behavior is
not captured by traditional perturbation theory. Furthermore, we analyze the
phonon self-energy and find that anharmonicity mixes guest rattling modes and
cage acoustic modes, significantly changing the character of the harmonic
phonons.
