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The initial mass function (IMF) is a key ingredient in many studies of galaxyformation and evolution. Although the IMF is often assumed to be universal,there is continuing evidence that it is not universal. Spectroscopic studiesthat derive the IMF of the unresolved stellar populations of a galaxy oftenassume that this spectrum can be described by a single stellar population(SSP). To alleviate these limitations, in this paper we have developed a uniquehierarchical Bayesian framework for modelling composite stellar populations(CSPs). Within this framework we use a parameterized IMF prior to regulate adirect inference of the IMF. We use this new framework to determine the numberof SSPs that is required to fit a set of realistic CSP mock spectra. The CSPmock spectra that we use are based on semi-analytic models and have an IMF thatvaries as a function of stellar velocity dispersion of the galaxy. Our resultssuggest that using a single SSP biases the determination of the IMF slope to ahigher value than the true slope, although the trend with stellar velocitydispersion is overall recovered. If we include more SSPs in the fit, theBayesian evidence increases significantly and the inferred IMF slopes of ourmock spectra converge, within the errors, to their true values. Most of thebias is already removed by using two SSPs instead of one. We show that we canreconstruct the variable IMF of our mock spectra for signal-to-noise ratiosexceeding $\sim$75.

Hierarchical Bayesian inference of the Initial Mass Function in Composite Stellar Populations