  We present a new flexible Bayesian framework for directly inferring the
fraction of neutral hydrogen in the intergalactic medium (IGM) during the Epoch
of Reionization (EoR, z~6-10) from detections and non-detections of Lyman Alpha
(Ly$\alpha$) emission from Lyman break galaxies (LBGs). Our framework combines
sophisticated reionization simulations with empirical models of the
interstellar medium (ISM) radiative transfer effects on Ly$\alpha$. We assert
that the Ly$\alpha$ line profile emerging from the ISM has an important impact
on the resulting transmission of photons through the IGM, and that these line
profiles depend on galaxy properties. We model this effect by considering the
peak velocity offset of Ly$\alpha$ lines from host galaxies' systemic
redshifts, which are empirically correlated with UV luminosity and redshift (or
halo mass at fixed redshift). We use our framework on the sample of LBGs
presented in Pentericci et al. (2014) and infer a global neutral fraction at
z~7 of $\overline{x}_\mathrm{HI} = 0.59_{-0.15}^{+0.11}$, consistent with other
robust probes of the EoR and confirming reionization is on-going ~700 Myr after
the Big Bang. We show that using the full distribution of Ly$\alpha$ equivalent
width detections and upper limits from LBGs places tighter constraints on the
evolving IGM than the standard Ly$\alpha$ emitter fraction, and that larger
samples are within reach of deep spectroscopic surveys of gravitationally
lensed fields and JWST NIRSpec.
