Not yet reviewed by Pith; the record is open.

The $^3{\rm H}(d,n)^4{\rm He}$ reaction is of significant interest in nuclear astrophysics and nuclear applications. It is an important, early step in big-bang nucleosynthesis and a key process in nuclear fusion reactors. We use one- and two-level $R$-matrix approximations to analyze data on the cross section for this reaction at center-of-mass energies below 215 keV. We critically examine the data sets using a Bayesian statistical model that allows for both common-mode and additional point-to-point uncertainties. We use Markov Chain Monte Carlo sampling to evaluate this $R$-matrix-plus-statistical model and find two-level $R$-matrix results that are stable with respect to variations in the channel radii. The $S$ factor at 40 keV evaluates to $25.36(19)$ MeV b (68% credibility interval). We discuss our Bayesian analysis in detail and provide guidance for future applications of Bayesian methods to $R$-matrix analyses. We also discuss possible paths to further reduction of the $S$-factor uncertainty.