The ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction governs the carbon-to-oxygen ratio set during helium burning, shaping white-dwarf structure and Type Ia supernova yields. At the astrophysical energy $E\approx 300\mathrm{keV}$, the cross section is controlled by the subthreshold $1^{-}$ (7.12 MeV) and $2^{+}$ (6.92 MeV) states, whose contributions depend on their asymptotic normalization coefficients (ANCs) $C_1$ and $C_2$, respectively. I perform a Bayesian analysis of the $S_{E1}$ and $S_{E2}$ factors using calibrated $R$-matrix mappings and contemporary ANC constraints for the $1^{-}, 2^{+}$, and $0^{+}$ ground state. For $S_{E1}$, a flat prior on the $1^{-}$ ANC yields a broad posterior (MAP $\approx 95.5\mathrm{keV}\mathrm{b}$), while Gaussian priors centered at the low and high anchors produce narrower posteriors with MAPs $\approx 75.4$ and $\approx 93.2\mathrm{keV}\mathrm{b}$, respectively. For $S_{E2}$, I employ a calibrated quadratic map in $(C_2,C_0)$ that captures the interference between the subthreshold $2^{+}$ capture and the external direct capture; the resulting posteriors reflect how the joint $(C_2,C_0)$ posterior loads the interference corridor. The resulting $S\left(300\mathrm{keV}\right)$ range is consistent with stellar-evolution models that produce low-spin, high-mass ($\approx 50-70M_{\odot }$) black holes. Overall, the Bayesian framework yields statistically robust posteriors for $S_{E1}$ and $S_{E2}$, quantifying correlations and uncertainties relevant for stellar modeling. This work links the microscopic subthreshold ANCs ($C_1,C_2,C_0$) to a macroscopic astrophysical outcome—the formation of massive, low-spin black holes—thereby connecting nuclear reaction experiments and theory with stellar evolution, black-hole demographics, and gravitational-wave observations.

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• H & He induced nuclear reactions
• Nuclear astrophysics
• Radiative capture
• S-factor
• 6 ≤ A ≤ 19
• Bayesian methods