  The Hubble Frontier Fields represent the opportunity to probe the
high-redshift evolution of the main sequence of star-forming galaxies to lower
masses than possible in blank fields thanks to foreground lensing of massive
galaxy clusters. We use the BEAGLE SED-fitting code to derive stellar masses,
$\mathrm{M_{\star}}=\log(M/\mathrm{M_{\odot}})$, SFRs,
$\Psi=\log(\psi/\mathrm{M_{\odot}}\,\mathrm{yr}^{-1})$ and redshifts from
galaxies within the ASTRODEEP catalogue. We fit a fully Bayesian hierarchical
model of the main sequence over $1.25<z<6$ of the form $\Psi =
\alpha_\mathrm{9.7}(z) + \beta(\mathrm{M_{\star}}-9.7) +
\mathcal{N}(0,\sigma^2)$ while explicitly modelling the outlier distribution.
The redshift-dependent intercept at $\mathrm{M_{\star}}=9.7$ is parametrized as
$\alpha_\mathrm{9.7}(z) = \log[N (1+z)^{\gamma}] + 0.7$. Our results agree with
an increase in normalization of the main sequence to high redshifts that
follows the redshift-dependent rate of accretion of gas onto dark matter halos
with $\gamma=2.40^{+0.18}_{-0.18}$. We measure a slope and intrinsic scatter of
$\beta=0.79^{+0.03}_{-0.04}$ and $\sigma=0.26^{+0.02}_{-0.02}$. We find that
the sampling of the SED provided by the combination of filters (Hubble +
ground-based Ks-band + Spitzer 3.6 and 4.5 $\mathrm{\mu m}$) is insufficient to
constrain $\mathrm{M_{\star}}$ and $\Psi$ over the full dynamic range of the
observed main sequence, even at the lowest redshifts studied. While this filter
set represents the best current sampling of high-redshift galaxy SEDs out to
$z>3$, measurements of the main sequence to low masses and high redshifts still
strongly depend on priors employed in SED fitting (as well as other fitting
assumptions). Future data-sets with JWST should improve this.
