Table B2.

Model comparison for joint data sets.

ModelVariantLFHFJ21F23LFHFJ21F23
  |$\Delta \log (Z)$||$\Delta \log (Z)$||$\Delta \log (Z)$||$\Delta \log (Z)$|C/DOFC/DOFC/DOFC/DOF
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
A 608.81309.6160.4268.45936/46748266/57643019/25933056/2671
B 158.875.262.789.65247/46726372/57622747/25912776/2669
C 12.83.314.513.64948/46716218/57612640/25902619/2668
C2Cut-off PL1.90.6|$\bigstar$||$\bigstar$|4935/46706215/57602610/25892589/2667
C3nthcomp cont.9.810.84.17.94936/46706231/57602609/25892596/2667
C4Torus2.60.42.45.04942/46696197/57592622/25882600/2666
D|$_n$| 13.918.012.910.84933/46706278/57602631/25892604/2667
D|$_i$|CF = 131.936.415.217.74985/46696245/57592693/25882641/2666
D|$_i$|CF = 0.6631.925.212.617.74985/46696444/57592751/25882634/2666
D|$_i$|CF = 0.334.935.411.60.14998/46696438/57592656/25882637/2666
E|$\log \xi =0$|⁠, |$a_*=0$|100.4208.546.158.65192/46726680/57612719/25902695/2668
E|$\log \xi =1$|⁠, |$a_*=0$|99.2202.043.655.85129/46726656/57612702/25902691/2668
E|$\log \xi =2$|⁠, |$a_*=0$|89.0188.041.652.65105/46727759/56712702/25902685/2668
E|$\log \xi =3$|⁠, |$a_*=0$|83.6194.440.455.15094/46726618/57612690/25902680/2668
E|$\log \xi =4$|⁠, |$a_*=0$|116.8200.251.475.35164/46726751/57612708/25902700/2668
E|$\log \xi =0$|⁠, |$a_*=0.98$|101.8208.447.059.45154/46726850/57612741/25902710/2668
F|$\log \xi =0$|⁠, |$a_*=0$|52.852.839.640.15016/46706261/57602664/25892642/2667
F|$\log \xi =1$|⁠, |$a_*=0$|55.729.138.540.95022/46706266/57602664/25892643/2667
F|$\log \xi =2$|⁠, |$a_*=0$|59.239.738.843.65026/46706286/57602662/25892653/2667
F|$\log \xi =3$|⁠, |$a_*=0$|80.529.543.457.35055/46706266/56702674/25892672/2667
F|$\log \xi =4$|⁠, |$a_*=0$|114.133.237.856.75141/46706283/57602695/25892689/2667
F|$\log \xi =0$|⁠, |$a_*=0.98$|56.631.241.042.75023/46706269/57602666/25892645/2667
F|$\log \xi =3$|⁠, |$a_*=0.98$|81.833.043.957.85055/45706272/57602675/25892672/2667
G|$\log \xi =0$|⁠, |$a_*=0$|3.0|$\bigstar$|12.78.64918/46686209/57592610/25882586/2666
G|$\log \xi =1$|⁠, |$a_*=0$|22.939.97.19.24923/46686210/57592610/25882586/2666
G|$\log \xi =2$|⁠, |$a_*=0$|14.238.44.76.34918/46686210/57592608/25882584/2666
G|$\log \xi =3$|⁠, |$a_*=0$|0.97.32.52.94913/46686206/57592608/25882585/2666
G|$\log \xi =4$|⁠, |$a_*=0$|16.839.923.26.94921/46686209/57592610/25882586/2666
G|$\log \xi =0$|⁠, |$a_*=0.98$|21.039.77.49.44921/46686210/57592611/25882586/2666
G|$\log \xi =1$|⁠, |$a_*=0.98$|23.340.87.49.74923/46686210/57592610/25882585/2666
G|$\log \xi =2$|⁠, |$a_*=0.98$|15.238.74.96.44919/46686211/57592609/25882584/2666
G|$\log \xi =3$|⁠, |$a_*=0.98$||$\bigstar$|9.22.22.64912/46686207/57592608/25882585/2666
G|$\log \xi =4$|⁠, |$a_*=0.98$|17.140.16.66.84921/46686209/57592610/25882586/2666
H|$\log \xi =0$|⁠, |$a_*=0$|79.8182.427.438.35254/46707302/57602758/25892792/2667
H|$\log \xi =1$|⁠, |$a_*=0$|74.0208.929.840.55037/46706666/57602670/25892637/2667
H|$\log \xi =2$|⁠, |$a_*=0$|57.9208.124.336.25046/46706618/57602679/25892633/2667
H|$\log \xi =3$|⁠, |$a_*=0$|52.7201.719.236.04994/46706603/57602659/25892633/2667
H|$\log \xi =4$|⁠, |$a_*=0$|75.6307.929.048.05021/46706776/57602681/25892652/2667
ModelVariantLFHFJ21F23LFHFJ21F23
  |$\Delta \log (Z)$||$\Delta \log (Z)$||$\Delta \log (Z)$||$\Delta \log (Z)$|C/DOFC/DOFC/DOFC/DOF
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
A 608.81309.6160.4268.45936/46748266/57643019/25933056/2671
B 158.875.262.789.65247/46726372/57622747/25912776/2669
C 12.83.314.513.64948/46716218/57612640/25902619/2668
C2Cut-off PL1.90.6|$\bigstar$||$\bigstar$|4935/46706215/57602610/25892589/2667
C3nthcomp cont.9.810.84.17.94936/46706231/57602609/25892596/2667
C4Torus2.60.42.45.04942/46696197/57592622/25882600/2666
D|$_n$| 13.918.012.910.84933/46706278/57602631/25892604/2667
D|$_i$|CF = 131.936.415.217.74985/46696245/57592693/25882641/2666
D|$_i$|CF = 0.6631.925.212.617.74985/46696444/57592751/25882634/2666
D|$_i$|CF = 0.334.935.411.60.14998/46696438/57592656/25882637/2666
E|$\log \xi =0$|⁠, |$a_*=0$|100.4208.546.158.65192/46726680/57612719/25902695/2668
E|$\log \xi =1$|⁠, |$a_*=0$|99.2202.043.655.85129/46726656/57612702/25902691/2668
E|$\log \xi =2$|⁠, |$a_*=0$|89.0188.041.652.65105/46727759/56712702/25902685/2668
E|$\log \xi =3$|⁠, |$a_*=0$|83.6194.440.455.15094/46726618/57612690/25902680/2668
E|$\log \xi =4$|⁠, |$a_*=0$|116.8200.251.475.35164/46726751/57612708/25902700/2668
E|$\log \xi =0$|⁠, |$a_*=0.98$|101.8208.447.059.45154/46726850/57612741/25902710/2668
F|$\log \xi =0$|⁠, |$a_*=0$|52.852.839.640.15016/46706261/57602664/25892642/2667
F|$\log \xi =1$|⁠, |$a_*=0$|55.729.138.540.95022/46706266/57602664/25892643/2667
F|$\log \xi =2$|⁠, |$a_*=0$|59.239.738.843.65026/46706286/57602662/25892653/2667
F|$\log \xi =3$|⁠, |$a_*=0$|80.529.543.457.35055/46706266/56702674/25892672/2667
F|$\log \xi =4$|⁠, |$a_*=0$|114.133.237.856.75141/46706283/57602695/25892689/2667
F|$\log \xi =0$|⁠, |$a_*=0.98$|56.631.241.042.75023/46706269/57602666/25892645/2667
F|$\log \xi =3$|⁠, |$a_*=0.98$|81.833.043.957.85055/45706272/57602675/25892672/2667
G|$\log \xi =0$|⁠, |$a_*=0$|3.0|$\bigstar$|12.78.64918/46686209/57592610/25882586/2666
G|$\log \xi =1$|⁠, |$a_*=0$|22.939.97.19.24923/46686210/57592610/25882586/2666
G|$\log \xi =2$|⁠, |$a_*=0$|14.238.44.76.34918/46686210/57592608/25882584/2666
G|$\log \xi =3$|⁠, |$a_*=0$|0.97.32.52.94913/46686206/57592608/25882585/2666
G|$\log \xi =4$|⁠, |$a_*=0$|16.839.923.26.94921/46686209/57592610/25882586/2666
G|$\log \xi =0$|⁠, |$a_*=0.98$|21.039.77.49.44921/46686210/57592611/25882586/2666
G|$\log \xi =1$|⁠, |$a_*=0.98$|23.340.87.49.74923/46686210/57592610/25882585/2666
G|$\log \xi =2$|⁠, |$a_*=0.98$|15.238.74.96.44919/46686211/57592609/25882584/2666
G|$\log \xi =3$|⁠, |$a_*=0.98$||$\bigstar$|9.22.22.64912/46686207/57592608/25882585/2666
G|$\log \xi =4$|⁠, |$a_*=0.98$|17.140.16.66.84921/46686209/57592610/25882586/2666
H|$\log \xi =0$|⁠, |$a_*=0$|79.8182.427.438.35254/46707302/57602758/25892792/2667
H|$\log \xi =1$|⁠, |$a_*=0$|74.0208.929.840.55037/46706666/57602670/25892637/2667
H|$\log \xi =2$|⁠, |$a_*=0$|57.9208.124.336.25046/46706618/57602679/25892633/2667
H|$\log \xi =3$|⁠, |$a_*=0$|52.7201.719.236.04994/46706603/57602659/25892633/2667
H|$\log \xi =4$|⁠, |$a_*=0$|75.6307.929.048.05021/46706776/57602681/25892652/2667

Note. Comparison of Bayesian evidence Z (columns 3–6) for different models (column 1) fitted to our joint data sets (3.3). Models A through H are described in Appendix B3. For models where we ‘step through’ discrete values of certain parameters, the relevant values are listed in column (2). The evidence (i.e. the marginal likelihood) is calculated using the BXA software (Buchner et al. 2014), over the parameter space defined by the bounds in Table B1. The model with the highest evidence, for a given data set, is indicated with a star symbol (⁠|$\bigstar$|⁠). For all other models, we tabulate the difference in log-evidence, |$\Delta \log (Z)=\log (Z_{\mathrm{best}})-\log (Z_{\mathrm{model}})$|⁠, between that model and the best (⁠|$\bigstar$|⁠) model. Models with the difference in log-evidence listed in bold text have |$\Delta \log (Z)\lt 3$|⁠; adopting a rather conservative threshold (Section 3.3.2), we do not consider such models to be decisively disfavoured compared to the best model. We also tabulate the Cash statistic and degrees of freedom (C/DOF, columns 7–10) for an x spec C-stat optimization, initiated at the posterior peak-likelihood parameter values obtained during the relevant BXA run, to provide an idea of the goodness of fit for that model.

Table B2.

Model comparison for joint data sets.

ModelVariantLFHFJ21F23LFHFJ21F23
  |$\Delta \log (Z)$||$\Delta \log (Z)$||$\Delta \log (Z)$||$\Delta \log (Z)$|C/DOFC/DOFC/DOFC/DOF
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
A 608.81309.6160.4268.45936/46748266/57643019/25933056/2671
B 158.875.262.789.65247/46726372/57622747/25912776/2669
C 12.83.314.513.64948/46716218/57612640/25902619/2668
C2Cut-off PL1.90.6|$\bigstar$||$\bigstar$|4935/46706215/57602610/25892589/2667
C3nthcomp cont.9.810.84.17.94936/46706231/57602609/25892596/2667
C4Torus2.60.42.45.04942/46696197/57592622/25882600/2666
D|$_n$| 13.918.012.910.84933/46706278/57602631/25892604/2667
D|$_i$|CF = 131.936.415.217.74985/46696245/57592693/25882641/2666
D|$_i$|CF = 0.6631.925.212.617.74985/46696444/57592751/25882634/2666
D|$_i$|CF = 0.334.935.411.60.14998/46696438/57592656/25882637/2666
E|$\log \xi =0$|⁠, |$a_*=0$|100.4208.546.158.65192/46726680/57612719/25902695/2668
E|$\log \xi =1$|⁠, |$a_*=0$|99.2202.043.655.85129/46726656/57612702/25902691/2668
E|$\log \xi =2$|⁠, |$a_*=0$|89.0188.041.652.65105/46727759/56712702/25902685/2668
E|$\log \xi =3$|⁠, |$a_*=0$|83.6194.440.455.15094/46726618/57612690/25902680/2668
E|$\log \xi =4$|⁠, |$a_*=0$|116.8200.251.475.35164/46726751/57612708/25902700/2668
E|$\log \xi =0$|⁠, |$a_*=0.98$|101.8208.447.059.45154/46726850/57612741/25902710/2668
F|$\log \xi =0$|⁠, |$a_*=0$|52.852.839.640.15016/46706261/57602664/25892642/2667
F|$\log \xi =1$|⁠, |$a_*=0$|55.729.138.540.95022/46706266/57602664/25892643/2667
F|$\log \xi =2$|⁠, |$a_*=0$|59.239.738.843.65026/46706286/57602662/25892653/2667
F|$\log \xi =3$|⁠, |$a_*=0$|80.529.543.457.35055/46706266/56702674/25892672/2667
F|$\log \xi =4$|⁠, |$a_*=0$|114.133.237.856.75141/46706283/57602695/25892689/2667
F|$\log \xi =0$|⁠, |$a_*=0.98$|56.631.241.042.75023/46706269/57602666/25892645/2667
F|$\log \xi =3$|⁠, |$a_*=0.98$|81.833.043.957.85055/45706272/57602675/25892672/2667
G|$\log \xi =0$|⁠, |$a_*=0$|3.0|$\bigstar$|12.78.64918/46686209/57592610/25882586/2666
G|$\log \xi =1$|⁠, |$a_*=0$|22.939.97.19.24923/46686210/57592610/25882586/2666
G|$\log \xi =2$|⁠, |$a_*=0$|14.238.44.76.34918/46686210/57592608/25882584/2666
G|$\log \xi =3$|⁠, |$a_*=0$|0.97.32.52.94913/46686206/57592608/25882585/2666
G|$\log \xi =4$|⁠, |$a_*=0$|16.839.923.26.94921/46686209/57592610/25882586/2666
G|$\log \xi =0$|⁠, |$a_*=0.98$|21.039.77.49.44921/46686210/57592611/25882586/2666
G|$\log \xi =1$|⁠, |$a_*=0.98$|23.340.87.49.74923/46686210/57592610/25882585/2666
G|$\log \xi =2$|⁠, |$a_*=0.98$|15.238.74.96.44919/46686211/57592609/25882584/2666
G|$\log \xi =3$|⁠, |$a_*=0.98$||$\bigstar$|9.22.22.64912/46686207/57592608/25882585/2666
G|$\log \xi =4$|⁠, |$a_*=0.98$|17.140.16.66.84921/46686209/57592610/25882586/2666
H|$\log \xi =0$|⁠, |$a_*=0$|79.8182.427.438.35254/46707302/57602758/25892792/2667
H|$\log \xi =1$|⁠, |$a_*=0$|74.0208.929.840.55037/46706666/57602670/25892637/2667
H|$\log \xi =2$|⁠, |$a_*=0$|57.9208.124.336.25046/46706618/57602679/25892633/2667
H|$\log \xi =3$|⁠, |$a_*=0$|52.7201.719.236.04994/46706603/57602659/25892633/2667
H|$\log \xi =4$|⁠, |$a_*=0$|75.6307.929.048.05021/46706776/57602681/25892652/2667
ModelVariantLFHFJ21F23LFHFJ21F23
  |$\Delta \log (Z)$||$\Delta \log (Z)$||$\Delta \log (Z)$||$\Delta \log (Z)$|C/DOFC/DOFC/DOFC/DOF
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
A 608.81309.6160.4268.45936/46748266/57643019/25933056/2671
B 158.875.262.789.65247/46726372/57622747/25912776/2669
C 12.83.314.513.64948/46716218/57612640/25902619/2668
C2Cut-off PL1.90.6|$\bigstar$||$\bigstar$|4935/46706215/57602610/25892589/2667
C3nthcomp cont.9.810.84.17.94936/46706231/57602609/25892596/2667
C4Torus2.60.42.45.04942/46696197/57592622/25882600/2666
D|$_n$| 13.918.012.910.84933/46706278/57602631/25892604/2667
D|$_i$|CF = 131.936.415.217.74985/46696245/57592693/25882641/2666
D|$_i$|CF = 0.6631.925.212.617.74985/46696444/57592751/25882634/2666
D|$_i$|CF = 0.334.935.411.60.14998/46696438/57592656/25882637/2666
E|$\log \xi =0$|⁠, |$a_*=0$|100.4208.546.158.65192/46726680/57612719/25902695/2668
E|$\log \xi =1$|⁠, |$a_*=0$|99.2202.043.655.85129/46726656/57612702/25902691/2668
E|$\log \xi =2$|⁠, |$a_*=0$|89.0188.041.652.65105/46727759/56712702/25902685/2668
E|$\log \xi =3$|⁠, |$a_*=0$|83.6194.440.455.15094/46726618/57612690/25902680/2668
E|$\log \xi =4$|⁠, |$a_*=0$|116.8200.251.475.35164/46726751/57612708/25902700/2668
E|$\log \xi =0$|⁠, |$a_*=0.98$|101.8208.447.059.45154/46726850/57612741/25902710/2668
F|$\log \xi =0$|⁠, |$a_*=0$|52.852.839.640.15016/46706261/57602664/25892642/2667
F|$\log \xi =1$|⁠, |$a_*=0$|55.729.138.540.95022/46706266/57602664/25892643/2667
F|$\log \xi =2$|⁠, |$a_*=0$|59.239.738.843.65026/46706286/57602662/25892653/2667
F|$\log \xi =3$|⁠, |$a_*=0$|80.529.543.457.35055/46706266/56702674/25892672/2667
F|$\log \xi =4$|⁠, |$a_*=0$|114.133.237.856.75141/46706283/57602695/25892689/2667
F|$\log \xi =0$|⁠, |$a_*=0.98$|56.631.241.042.75023/46706269/57602666/25892645/2667
F|$\log \xi =3$|⁠, |$a_*=0.98$|81.833.043.957.85055/45706272/57602675/25892672/2667
G|$\log \xi =0$|⁠, |$a_*=0$|3.0|$\bigstar$|12.78.64918/46686209/57592610/25882586/2666
G|$\log \xi =1$|⁠, |$a_*=0$|22.939.97.19.24923/46686210/57592610/25882586/2666
G|$\log \xi =2$|⁠, |$a_*=0$|14.238.44.76.34918/46686210/57592608/25882584/2666
G|$\log \xi =3$|⁠, |$a_*=0$|0.97.32.52.94913/46686206/57592608/25882585/2666
G|$\log \xi =4$|⁠, |$a_*=0$|16.839.923.26.94921/46686209/57592610/25882586/2666
G|$\log \xi =0$|⁠, |$a_*=0.98$|21.039.77.49.44921/46686210/57592611/25882586/2666
G|$\log \xi =1$|⁠, |$a_*=0.98$|23.340.87.49.74923/46686210/57592610/25882585/2666
G|$\log \xi =2$|⁠, |$a_*=0.98$|15.238.74.96.44919/46686211/57592609/25882584/2666
G|$\log \xi =3$|⁠, |$a_*=0.98$||$\bigstar$|9.22.22.64912/46686207/57592608/25882585/2666
G|$\log \xi =4$|⁠, |$a_*=0.98$|17.140.16.66.84921/46686209/57592610/25882586/2666
H|$\log \xi =0$|⁠, |$a_*=0$|79.8182.427.438.35254/46707302/57602758/25892792/2667
H|$\log \xi =1$|⁠, |$a_*=0$|74.0208.929.840.55037/46706666/57602670/25892637/2667
H|$\log \xi =2$|⁠, |$a_*=0$|57.9208.124.336.25046/46706618/57602679/25892633/2667
H|$\log \xi =3$|⁠, |$a_*=0$|52.7201.719.236.04994/46706603/57602659/25892633/2667
H|$\log \xi =4$|⁠, |$a_*=0$|75.6307.929.048.05021/46706776/57602681/25892652/2667

Note. Comparison of Bayesian evidence Z (columns 3–6) for different models (column 1) fitted to our joint data sets (3.3). Models A through H are described in Appendix B3. For models where we ‘step through’ discrete values of certain parameters, the relevant values are listed in column (2). The evidence (i.e. the marginal likelihood) is calculated using the BXA software (Buchner et al. 2014), over the parameter space defined by the bounds in Table B1. The model with the highest evidence, for a given data set, is indicated with a star symbol (⁠|$\bigstar$|⁠). For all other models, we tabulate the difference in log-evidence, |$\Delta \log (Z)=\log (Z_{\mathrm{best}})-\log (Z_{\mathrm{model}})$|⁠, between that model and the best (⁠|$\bigstar$|⁠) model. Models with the difference in log-evidence listed in bold text have |$\Delta \log (Z)\lt 3$|⁠; adopting a rather conservative threshold (Section 3.3.2), we do not consider such models to be decisively disfavoured compared to the best model. We also tabulate the Cash statistic and degrees of freedom (C/DOF, columns 7–10) for an x spec C-stat optimization, initiated at the posterior peak-likelihood parameter values obtained during the relevant BXA run, to provide an idea of the goodness of fit for that model.

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