Ultracold atomic gases in optical lattices have proven to be a controllable, tunable and clean implementation of strongly interacting quantum many-body systems. An essential prospect for such quantum simulators is their ability to map out the phase diagram of fundamental many-body model Hamiltonians. However, the results need to be validated first for representative benchmark problems through state-of-the-art numerical methods of quantum many-body theory. Here we present the first ab initio comparison between experiments and quantum Monte Carlo simulations for strongly interacting Bose gases on a lattice for large systems (up to particles). The comparison enables thermometry for the interacting quantum gas and to experimentally determine the finite-temperature phase diagram for bosonic superfluids in an optical lattice, revealing a suppression of the critical temperature as the transition to the Mott insulator is approached.