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We present a probabilistic approach for inferring the parameters of thepresent day power-law stellar mass function (MF) of a resolved young starcluster. This technique (a) fully exploits the information content of a givendataset; (b) accounts for observational uncertainties in a straightforward way;(c) assigns meaningful uncertainties to the inferred parameters; (d) avoids thepitfalls associated with binning data; and (e) is applicable to virtually anyresolved young cluster, laying the groundwork for a systematic study of thehigh mass stellar MF (M > 1 Msun). Using simulated clusters and Markov chainMonte Carlo sampling of the probability distribution functions, we show thatestimates of the MF slope, {\alpha}, are unbiased and that the uncertainty,{\Delta}{\alpha}, depends primarily on the number of observed stars and stellarmass range they span, assuming that the uncertainties on individual masses andthe completeness are well-characterized. Using idealized mock data, we computethe 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 thetheoretical lower limit. By correcting these uncertainties to the theoreticallower limits, we find the literature studies yield <{\alpha}>=2.46 with a1-{\sigma} dispersion of 0.35 dex. We verify that it is impossible for apower-law MF to obtain meaningful constraints on the upper mass limit of theIMF. 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) fullycharacterizing and then explicitly modeling the completeness for stars of agiven mass. (abridged)

astrophysical journal/the astrophysical journal

The Panchromatic Hubble Andromeda Treasury IV. A Probabilistic Approach  to Inferring the High Mass Stellar Initial Mass Function and Other Power-law  Functions