  Phenomenological $R$-matrix has been a standard framework for the evaluation
of resolved resonance cross section data in nuclear physics for many years. It
is a powerful method for comparing different types of experimental nuclear data
and combining the results of many different experimental measurements in order
to gain a better estimation of the true underlying cross sections. Yet a
practical challenge has always been the estimation of the uncertainty on both
the cross sections at the energies of interest and the fit parameters, which
can take the form of standard level parameters. Frequentist ($\chi^2$-based)
estimation has been the norm. In this work, a Markov Chain Monte Carlo sampler,
\texttt{emcee}, has been implemented for the $R$-matrix code \texttt{AZURE2},
creating the Bayesian $R$-matrix Inference Code Kit (\texttt{BRICK}). Bayesian
uncertainty estimation has then been carried out for a simultaneous $R$-matrix
fit of the $^3$He$(\alpha,\gamma)^7$Be and $^3$He$(\alpha,\alpha)^3$He
reactions in order to gain further insight into the fitting of capture and
scattering data. Both data sets constrain the values of the bound state
$\alpha$-particle asymptotic normalization coefficients in $^7$Be. The analysis
highlights the need for low-energy scattering data with well-documented
uncertainty information and shows how misleading results can be obtained in its
absence.
