  The intrinsically anti-ferromagnetic topological insulator, MnBi$_2$Te$_4$
(MBT), has garnered significant attention recently. The excitement for this
layered van der Waals bonded compound stems from its potential to host numerous
exotic topological quantum states. For instance, quantum anomalous Hall states
are predicted for odd-layer compounds and axion insulator states for the
even-layer compounds. Unfortunately, the realization of these phenomena has
been hindered by experimental challenges such as the existence of negative
charge carriers, i.e., electron doping, which have been linked to anti-site
defects among the Mn and Bi sub-lattices. Based on high level diffusion Monte
Carlo (DMC) and DMC-tuned DFT+U calculations, we provide benchmark quality
results for the bulk Mn magnetization as well as for Mn$_{Bi}$ and Bi$_{Mn}$
defects. We use this information to refine and extend models that estimate the
anti-site defect concentration in actual MBT samples when combined with data
from magnetic susceptibility and intermediate field magnetization measurements.
Our models are validated through favorable comparison with prior experimental
studies that obtained both magnetic and site occupancy data. We then extend our
estimates to a larger set of prior samples to identify a probable zone of low
defect density that has yet to be reached in synthesis. We anticipate our
theoretically based magnetic purity measures may be used as minimization
targets in the cycle of refinement needed to synthesize MBT samples with low
anti-site defect concentrations and more reproducible topological properties.
