All-inorganic lead halide perovskites $\mathrm{CsPb}{\mathrm{X}}_3$ (X = Cl, Br, I) exhibit rich structural dynamics, with vibrational properties and electron-lattice interactions still under active investigation due to strong lattice anharmonicity and structural phase transitions. Here we present a first-principles study of $\mathrm{CsPbB}{\mathrm{r}}_3$ across its orthorhombic, tetragonal, and cubic phases. We analyze the impact of the structural phases on the phonon dispersions, by including lattice anharmonicity. We discuss the validity of the semiclassical Fröhlich Hamiltonian, by quantifying the electron-phonon coupling strength for individual polar optical phonons, analyzing the mixing induced by long-range polar interactions and the relative contributions to the dielectric spectrum, tracing their symmetry evolution using group theory and a band folding approach. Despite pronounced structural transformations and enhanced anharmonicity at elevated temperatures, the long-range polar coupling mechanism associated with a single longitudinal optical mode lying at high frequency remains dominant throughout the entire temperature range. Furthermore, we show that, in the absence of symmetry-breaking effects, effective cubic models accurately capture long-range field behavior in low-temperature phases. This work establishes a microscopic foundation for a better understanding of polarons and exciton-polarons in bulk $\mathrm{CsPbB}{\mathrm{r}}_3$ and related nanostructures.

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• Anharmonic lattice dynamics
• Electron-phonon coupling
• Polarons
• Perovskites
• Density functional calculations