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

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.

Figure 1

Correlation between $E_0$ and $E_B$. Blue, shaded regions are the samples from the MCMC analysis. The solid, orange line is the predicted correlation according to Eq. (26). The green, dashed lines are the boundaries of the uniform prior applied to $E_0$ in Ref. [12].

Figure 2

Correlations between ${\gamma }_d^2$ and $a_d$ (top) and ${\gamma }_n^2$ and $a_n$ (bottom). Samples from the MCMC analysis are shown as blue circles. Solid, orange lines represent the correlation imposed by Eq. (28). Dashed, green lines represent the fitted results of Eq. (29).

Figure 3

$S\left(40\mathrm{keV}\right)$ posteriors for different statistical models. The $S\left(40\mathrm{keV}\right)$ posteriors were generated from an analysis with a single $R$-matrix level at fixed channel radii $a_d=5.56$ fm and $a_n=3.633$ fm. ${\mathcal{U}}_{af}$ (blue) is shown in the top panel in comparison to a summary of the result of [12] (purple). The posteriors for four different statistical models, ${\mathcal{U}}_{af}$ (blue), ${\mathcal{U}}_f$ (orange), ${\mathcal{U}}_a$ (green), and ${\mathcal{U}}_{rf}$ (red), are presented in the bottom panel for comparison.

Figure 4

$R$-matrix-parameter posteriors for a single-level $R$-matrix analysis at $a_d=5.56$ fm and $a_n=3.633$ fm combined with statistical model ${\mathcal{U}}_{af}$.

Figure 5

Summary of the absolute extrinsic uncertainty posteriors from an analysis using a single $R$-matrix level and channel radii of $a_d=5.56$ and $a_n=3.633$ fm. The lower and upper limits of the blue error bars correspond to 16% and 84% quantiles, respectively. For comparison, the same quantiles reported in [12] are shown in orange.

Figure 6

Comparison of the normalization factors obtained with a single-level $R$-matrix approximation and absolute extrinsic uncertainties. Our results are shown in blue. The results reported in [12] are shown in orange. The error bars indicate 16% and 84% quantiles.

Figure 7

$S\left(40\mathrm{keV}\right)$ (a) and $S\left(140\mathrm{keV}\right)$ (b) posteriors using two $3/2^{+}R$-matrix levels and statistical model ${\mathcal{U}}_{rf}$ for the analysis of all data sets, including that of Kobzev et al.. Posteriors are shown at $a_d=4.25$ (blue), 5.56 (orange), and 7.25 (green) fm. $a_n$ is fixed at 3.633 fm.

Figure 8

Cross section residual comparison of the Kobzev data when energies are sampled (orange squares) versus when they are not (blue circles).

Figure 9

Residual comparison of the Kobzev data. The cross section residuals are shown as blue circles, and the energy residuals are shown as orange squares.

Figure 10

Experimental cross sections above 100 keV (COM) relative to the theory prediction at $maxln\mathcal{L}$ for the Kobzev (blue circles) and Conner (orange squares) data sets.

Figure 11

$S\left(40\mathrm{keV}\right)$, $S\left(140\mathrm{keV}\right)$, and ${\alpha }_j$ dependence on $a_d$. Model A results are shown in the left column, panels (a), (c), and (e). Model B results are shown in the right column, panels (b), (d), and (f). Blue circles correspond to results obtained for the Jarmie et al. [4] (blue circles), Brown et al. [5] (orange squares), Arnold et al. [2] (green diamonds), and Conner et al. [1] (red stars) are shown individually in the bottom row. Error bars reflect the 16% and 84% quantiles.

Figure 12

The evolution of the product ${\mathrm{\Gamma }}_{2d}{\mathrm{\Gamma }}_{2n}$ is given for all $a_d$ values under consideration. Results were obtained with $R$-matrix Model B and statistical model ${\mathcal{U}}_{rf}$. $a_n$ was fixed at 3.633 fm.

Figure 13

Cross section data (without extrinsic uncertainties) from the four sets in our final analysis compared to a series of curves generated (without normalization factors) from a subset of the chains for each of the $a_d$ values in the first column of Table 2.

Figure 14

Residuals as defined by (36) for the four data sets included in our final evaluation at four different values of $a_d$: 4.25 (blue circles), 5.00 (orange squares), 5.56 (green diamonds), and 7.25 fm (red stars). Panel (a) corresponds to Jarmie et al. [4], (b) to Brown et al. [5], (c) to Arnold et al. [2], and (d) to Conner et al. [1]. The residuals for different channel radii are so consistent that all four symbols often lie essentially on top of each other. As indicated in the figure, panels (a)–(c) share the same $x$-axis scale while panel (d) extends over a wider energy range. Results were obtained using $R$-matrix Model B and statistical model ${\mathcal{U}}_{rf}$. $a_n$ was fixed at 3.633 fm.

Figure 15

$S\left(40\mathrm{keV}\right)$ posterior generated with $R$-matrix Model B and statistical model ${\mathcal{U}}_{rf}$. Results in $25.{36}_{-0.19}^{+0.19}$ MeV b. Since results from all $a_d$ values in the first column of Table 2 are statistically consistent they are combined to produce one final result.

Figure 16

$S\left(140\mathrm{keV}\right)$ posterior generated with $R$-matrix Model B and statistical model ${\mathcal{U}}_{rf}$. Results in $5.{12}_{-0.05}^{+0.04}$ MeV b. Results from all $a_d$ values in the first column of Table 2 are combined, as in Fig. 15.

Figure 17

$R$-matrix parameter posteriors for $a_d=7.25$ fm and $a_n=3.633$ fm using $R$-matrix Model B and statistical model ${\mathcal{U}}_{rf}$. $E_r$ and the partial widths are presented in MeV and $U_e$ in eV.

Figure 18

Relative extrinsic uncertainty parameter posteriors for $a_d=7.25$ fm and $a_n=3.633$ fm using $R$-matrix Model B and statistical model ${\mathcal{U}}_{rf}$.

Figure 19

Normalization factor parameter posteriors for $a_d=7.25$ fm and $a_n=3.633$ fm using $R$-matrix Model B and statistical model ${\mathcal{U}}_{rf}$. These normalization factors were applied to the theory predictions.