The Panchromatic Hubble Andromeda Treasury IV. A Probabilistic Approach
  to Inferring the High Mass Stellar Initial Mass Function and Other Power-law
  Functions
  We present a probabilistic approach for inferring the parameters of the
present day power-law stellar mass function (MF) of a resolved young star
cluster. This technique (a) fully exploits the information content of a given
dataset; (b) accounts for observational uncertainties in a straightforward way;
(c) assigns meaningful uncertainties to the inferred parameters; (d) avoids the
pitfalls associated with binning data; and (e) is applicable to virtually any
resolved young cluster, laying the groundwork for a systematic study of the
high mass stellar MF (M > 1 Msun). Using simulated clusters and Markov chain
Monte Carlo sampling of the probability distribution functions, we show that
estimates of the MF slope, {\alpha}, are unbiased and that the uncertainty,
{\Delta}{\alpha}, depends primarily on the number of observed stars and stellar
mass range they span, assuming that the uncertainties on individual masses and
the completeness are well-characterized. Using idealized mock data, we compute
the lower limit precision on {\alpha} and provide an analytic approximation for
{\Delta}{\alpha} as a function of the observed number of stars and mass range.
We find that ~ 3/4 of quoted literature uncertainties are smaller than the
theoretical lower limit. By correcting these uncertainties to the theoretical
lower limits, we find the literature studies yield <{\alpha}>=2.46 with a
1-{\sigma} dispersion of 0.35 dex. We verify that it is impossible for a
power-law MF to obtain meaningful constraints on the upper mass limit of the
IMF. We show that avoiding substantial biases in the MF slope requires: (1)
including the MF as a prior when deriving individual stellar mass estimates;
(2) modeling the uncertainties in the individual stellar masses; and (3) fully
characterizing and then explicitly modeling the completeness for stars of a
given mass. (abridged)
