  Accurate specification of a likelihood function is becoming increasingly
difficult in many inference problems in astronomy. As sample sizes resulting
from astronomical surveys continue to grow, deficiencies in the likelihood
function lead to larger biases in key parameter estimates. These deficiencies
result from the oversimplification of the physical processes that generated the
data, and from the failure to account for observational limitations.
Unfortunately, realistic models often do not yield an analytical form for the
likelihood. The estimation of a stellar initial mass function (IMF) is an
important example. The stellar IMF is the mass distribution of stars initially
formed in a given cluster of stars, a population which is not directly
observable due to stellar evolution and other disruptions and observational
limitations of the cluster. There are several difficulties with specifying a
likelihood in this setting since the physical processes and observational
challenges result in measurable masses that cannot legitimately be considered
independent draws from an IMF. This work improves inference of the IMF by using
an approximate Bayesian computation approach that both accounts for
observational and astrophysical effects and incorporates a physically-motivated
model for star cluster formation. The methodology is illustrated via a
simulation study, demonstrating that the proposed approach can recover the true
posterior in realistic situations, and applied to observations from
astrophysical simulation data.
