  Big bang nucleosynthesis provides the earliest probe of standard model
physics, at a time when the universe was less than a thousand seconds old. It
determines the abundances of the lightest nuclides, which give rise to the
subsequent history of the visible matter in the Universe. This work derives new
$^7$Be(n,p)$^7$Li thermonuclear reaction rates based on all available
experimental information. This reaction sensitively impacts the primordial
abundances of $^{7}$Be and $^7$Li during big bang nucleosynthesis. We
critically evaluate all available data and disregard experimental results that
are questionable. For the nuclear model, we adopt an incoherent sum of
single-level, two-channel R-matrix approximation expressions, which are
implemented into a hierarchical Bayesian model, to analyze the remaining six
data sets we deem most reliable. In the fitting of the data, we consistently
model all known sources of uncertainty, including discrepant absolute
normalizations of different data sets, and also take the variation of the
neutron and proton channel radii into account, hence providing less biased
estimates of the $^7$Be(n,p)$^7$Li thermonuclear rates. From the resulting
posteriors, we extract R-matrix parameters ($E_r$, $\gamma^2_n$, $\gamma^2_p$)
and derive excitation energies, partial and total widths. Our fit is sensitive
to the contributions of the first three levels above the neutron threshold.
Reaction rates were computed by integrating 10,000 samples of the reduced cross
section. Our $^7$Be(n,p)$^7$Li thermonuclear rates have uncertainties between
1.5% and 2.0% at temperatures of $\leq$1 GK. We compare our rates to previous
results and find that the $^7$Be(n,p)$^7$Li rates most commonly used in big
bang simulations have too optimistic uncertainties.
