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.
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