  We developed a hierarchical Bayesian framework to estimate S-factors and
thermonuclear rates for the $^3$He(d,p)$^{4}$He reaction, which impacts the
primordial abundances of $^3$He and $^7$Li. The available data are evaluated
and all direct measurements are taken into account in our analysis for which we
can estimate separate uncertainties for systematic and statistical effects. For
the nuclear reaction model, we adopt a single-level, two-channel approximation
of R-matrix theory, suitably modified to take the effects of electron screening
at lower energies into account. Apart from the usual resonance parameters
(resonance location and reduced widths for the incoming and outgoing reaction
channel), we include for the first time the channel radii and boundary
condition in the fitting process. Our new analysis of the $^3$He(d,p)$^{4}$He
S-factor data results in improved estimates for the thermonuclear rates. This
work represents the first nuclear rate evaluation using the R-matrix theory
embedded into a hierarchical Bayesian framework, properly accounting for all
known sources of uncertainty. Therefore, it provides a test bed for future
studies of more complex reactions.

Instrumentation and Methods for Astrophysics;Data Analysis, Statistics and Probability