İstanbul's Hagia Sophia is a monumental structure with multiple sub-spaces coupled to one another through arches. Its architectural elements have undergone alterations as its function has changed from that of a church to a mosque, a mosque to a museum, and back to a mosque. This study makes use of Hagia Sophia's rich formal and material characteristics to conduct a comprehensive investigation of room acoustics coupling. The methodology involves the application of the diffusion equation model (DEM) for sound energy flow analysis. Energy flow decays and energy flow dips are examined for almost 1000 receiver positions distributed throughout the various sub-spaces of the building. Ray-tracing (Ray-t) simulations are used to support the energy flow decay analysis conducted using DEM. The Ray-t data are subjected to Bayesian analysis to identify the decay parameters and the degree of acoustical coupling. Among the many variables, the source-receiver distance and positioning within different sub-spaces appear to be the underlying determinant of multi-slope sound decay pattern. On the other hand, the cases of multi-slope decays identified within the structure tend to weaken and single-slope cases increase when the overall absorption area increases in the mosque state due to the carpeted floor.