We present the implementation of a Bayesian formalism within the Stochastically Lighting Up Galaxies (SLUG) stellar population synthesis code, which is designed to investigate variations in the initial mass function (IMF) of star clusters. By comparing observed cluster photometry to large libraries of clusters simulated with a continuously varying IMF, our formalism yields the posterior probability distribution function (PDF) of the cluster mass, age, and extinction, jointly with the parameters describing the IMF. We apply this formalism to a sample of star clusters from the nearby galaxy NGC 628, for which broad-band photometry in five filters is available as part of the Legacy ExtraGalactic UV Survey (LEGUS). After allowing the upper-end slope of the IMF ($alpha_3$) to vary, we recover PDFs for the mass, age, and extinction that are broadly consistent with what is found when assuming an invariant Kroupa IMF. However, the posterior PDF for $alpha_3$ is very broad due to a strong degeneracy with the cluster mass, and it is found to be sensitive to the choice of priors, particularly on the cluster mass. We find only a modest improvement in the constraining power of $alpha_3$ when adding H$alpha$ photometry from the companion H$alpha$-LEGUS survey. Conversely, H$alpha$ photometry significantly improves the age determination, reducing the frequency of multi-modal PDFs. With the aid of mock clusters we quantify the degeneracy between physical parameters, showing how constraints on the cluster mass that are independent of photometry can be used to pin down the IMF properties of star clusters.
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