  We present a new analysis of the rest-frame UV and optical spectra of a
sample of three $z>8$ galaxies discovered behind the gravitational lensing
cluster RX\,J2129.4+0009. We combine these observations with $z>7.5$ galaxies
from the literature, for which similar measurements are available. As already
pointed out in other studies, the high [\oiii]$\lambda$5007/[\oii]$\lambda$3727
ratios ($O_{32}$) and steep UV continuum slopes ($\beta$) are consistent with
the values observed for low redshift Lyman continuum emitters, suggesting that
such galaxies contribute to the ionizing budget of the intergalactic medium. We
construct a logistic regression model to estimate the probability of a galaxy
being a Lyman continuum emitter based on the measured \MUV, $\beta$, and
$O_{32}$. Using this probability and the UV luminosity function, we construct
an empirical model that estimates the contribution of high redshift galaxies to
reionization. The preferred scenario in our analysis shows that at $z\sim8$,
the average escape fraction of the galaxy population (i.e., including both LyC
emitters and non-emitters) varies with \MUV, with intermediate UV luminosity
($-19<M_{UV}<-16$) galaxies having larger escape fraction. Galaxies with faint
UV luminosity ($-16<M_{UV}<-13.5$) contribute most of the ionizing photons. The
relative contribution of faint versus bright galaxies depends on redshift, with
the intermediate UV galaxies becoming more important over time. UV bright
galaxies, although more likely to be LCEs at a given log($O_{32}$) and $\beta$,
contribute the least of the total ionizing photon budget.
