  [Abbreviated] We present the largest search to date for $z\sim8$ Lyman break
galaxies (LBGs) based on 350 arcmin$^2$ of HST observations in the V-, Y-, J-
and H-bands from the Brightest of Reionizing Galaxies (BoRG) survey. The BoRG
dataset includes $\sim$50 arcmin$^2$ of new data and deeper observations of two
previous BoRG pointings, from which we present 9 new $z\sim8$ LBG candidates,
bringing the total number of BoRG LBGs to 38 with $25.5\leqslant m_{J}
\leqslant 27.6$ (AB system). We introduce a new Bayesian formalism for
estimating the galaxy luminosity function (LF), which does not require binning
(and thus smearing) of the data and includes a likelihood based on the formally
correct binomial distribution as opposed to the often used approximate Poisson
distribution. We demonstrate the utility of the new method on a sample of $97$
LBGs that combines the bright BoRG galaxies with the fainter sources published
in Bouwens et al. (2012) from the HUDF and ERS programs. We show that the
$z\sim8$ LF is well described by a Schechter function with a characteristic
magnitude $M^\star = -20.15^{+0.29}_{-0.38}$, a faint-end slope of $\alpha =
-1.87^{+0.26}_{-0.26}$, and a number density of $\log_{10} \phi^\star
[\textrm{Mpc}^{-3}] = -3.24^{+0.25}_{-0.24}$. Integrated down to $M=-17.7$ this
LF yields a luminosity density, $\log_{10} \epsilon
[\textrm{erg}/\textrm{s/Hz/Mpc}^{3}] = 25.52^{+0.05}_{-0.05}$. Our LF analysis
is consistent with previously published determinations within 1$\sigma$. We
discuss the implication of our study for the physics of reionization. By
assuming theoretically motivated priors on the clumping factor and the photon
escape fraction we show that the UV LF from galaxy samples down to $M=-17.7$
can ionize only 10-50% of the neutral hydrogen at $z\sim8$. Full reionization
would require extending the LF down to $M=-15$.
