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Abstract Measuring observables to constrain models using maximum-likelihood estimation is fundamental to many physics experiments. Wilks' theorem provides a simple way to construct confidence intervals on model parameters, but it only applies under certain conditions. These conditions, such as nested hypotheses and unbounded parameters, are often violated in neutrino oscillation measurements and other experimental scenarios. Monte Carlo methods can address these issues, albeit at increased computational cost. In the presence of nuisance parameters, however, the best way to implement a Monte Carlo method is ambiguous. This paper documents the method selected by the NOvA experiment, the profile construction. It presents the toy studies that informed the choice of method, details of its implementation, and tests performed to validate it. It also includes some practical considerations which may be of use to others choosing to use the profile construction. 

NOvA is a long-baseline neutrino oscillation experiment that measures oscillations in charged-current ${\nu }_{\mu }\to {\nu }_{\mu }$(disappearance) and ${\nu }_{\mu }\to {\nu }_e$(appearance) channels, and their antineutrino counterparts, using neutrinos of energies around 2 GeV over a distance of 810 km. In this work we reanalyze the dataset first examined in our previous paper [] using an alternative statistical approach based on Bayesian Markov chain Monte Carlo. We measure oscillation parameters consistent with the previous results. We also extend our inferences to include the first NOvA measurements of the reactor mixing angle ${\theta }_{13}$, where we find $0.071\le {\sin }^{2}2{\theta }_{13}\le 0.107$, and the Jarlskog invariant, where we observe no significant preference for the $CP$-conserving value $J=0$over values favoring $CP$violation. We use these results to examine the effects of constraints from short-baseline measurements of ${\theta }_{13}$using antineutrinos from nuclear reactors when making NOvA measurements of ${\theta }_{23}$. Our long-baseline measurement of ${\theta }_{13}$is shown to be consistent with the reactor measurements, supporting the general applicability and robustness of the Pontecorvo-Maki-Nakagawa-Sakata framework for neutrino oscillations. Published by the American Physical Society2024