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The $^3{\rm H}(d,n)^4{\rm He}$ reaction is of significant interest in nuclearastrophysics and nuclear applications. It is an important, early step inbig-bang nucleosynthesis and a key process in nuclear fusion reactors. We useone- and two-level $R$-matrix approximations to analyze data on the crosssection for this reaction at center-of-mass energies below 215 keV. Wecritically examine the data sets using a Bayesian statistical model that allowsfor both common-mode and additional point-to-point uncertainties. We use MarkovChain Monte Carlo sampling to evaluate this $R$-matrix-plus-statistical modeland find two-level $R$-matrix results that are stable with respect tovariations in the channel radii. The $S$ factor at 40 keV evaluates to$25.36(19)$ MeV b (68% credibility interval). We discuss our Bayesian analysisin 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.

How Bayesian methods can improve $R$-matrix analyses of data: the example of the $dt$ Reaction