  With recent advances in modelling stars using high-precision
asteroseismology, the systematic effects associated with our assumptions of
stellar helium abundance ($Y$) and the mixing-length theory parameter
($\alpha_\mathrm{MLT}$) are becoming more important. We apply a new method to
improve the inference of stellar parameters for a sample of Kepler dwarfs and
subgiants across a narrow mass range ($0.8 < M < 1.2\,\mathrm{M_\odot}$). In
this method, we include a statistical treatment of $Y$ and the
$\alpha_\mathrm{MLT}$. We develop a hierarchical Bayesian model to encode
information about the distribution of $Y$ and $\alpha_\mathrm{MLT}$ in the
population, fitting a linear helium enrichment law including an intrinsic
spread around this relation and normal distribution in $\alpha_\mathrm{MLT}$.
We test various levels of pooling parameters, with and without solar data as a
calibrator. When including the Sun as a star, we find the gradient for the
enrichment law, $\Delta Y / \Delta Z = 1.05^{+0.28}_{-0.25}$ and the mean
$\alpha_\mathrm{MLT}$ in the population, $\mu_\alpha = 1.90^{+0.10}_{-0.09}$.
While accounting for the uncertainty in $Y$ and $\alpha_\mathrm{MLT}$, we are
still able to report statistical uncertainties of 2.5 per cent in mass, 1.2 per
cent in radius, and 12 per cent in age. Our method can also be applied to
larger samples which will lead to improved constraints on both the population
level inference and the star-by-star fundamental parameters.
