  We build a rigorous statistical framework to provide constraints on the
chemical and dust evolution parameters for nearby late-type galaxies with a
wide range of gas fractions ($3\%<f_g<94\%$). A Bayesian Monte Carlo Markov
Chain framework provides statistical constraints on the parameters used in
chemical evolution models. Nearly a million one-zone chemical and dust
evolution models were compared to 340 galaxies. Relative probabilities were
calculated from the $\chi^2$ between data and models, marginalised over the
different time steps, galaxy masses and star formation histories. We applied
this method to find `best fitting' model parameters related to metallicity, and
subsequently fix these metal parameters to study the dust parameters. For the
metal parameters, a degeneracy was found between the choice of initial mass
function, supernova metal yield tables and outflow prescription. For the dust
parameters, the uncertainties on the best fit values are often large except for
the fraction of metals available for grain growth, which is well constrained.
We find a number of degeneracies between the dust parameters, limiting our
ability to discriminate between chemical models using observations only. For
example, we show that the low dust content of low-metallicity galaxies can be
resolved by either reducing the supernova dust yields and/or including
photo-fragmentation. We also show that supernova dust dominates the dust mass
for low metallicity galaxies and grain growth dominates for high metallicity
galaxies. The transition occurs around $12+\log({\rm O/H})=7.75$, which is
lower than found in most studies in the literature.
