Robust uncertainty quantification in popular estimators of the instantaneous reproduction number

The instantaneous reproduction number (Rt) is a key measure of the rate of spread of an infectious disease. Correctly quantifying uncertainty in Rt estimates is crucial for making well-informed decisions. Popular Rt estimators leverage smoothing techniques to distinguish signal from noise. Examples include EpiEstim and EpiFilter, which are both controlled by a "smoothing parameter" that is traditionally selected by users. We demonstrate that the values of these smoothing parameters are unknown, vary markedly with epidemic dynamics, and show that data-driven smoothing is crucial for accurate uncertainty quantification of Rt estimates. We derive model likelihoods for the smoothing parameters in both EpiEstim and EpiFilter and develop a Bayesian framework to automatically marginalise these parameters when fitting to epidemiological time-series data. This yields novel marginal posterior predictive distributions which prove integral to rigorous model evaluation. Applying our methods, we find that default parameterisations of these widely-used estimators can negatively impact Rt inference, delaying detection of epidemic growth, and misrepresenting uncertainty (typically producing overconfident estimates), with implications for public health decision-making. Our extensions mitigate these issues, provide a principled approach to uncertainty quantification, improve the robustness of real-time Rt inference, and facilitate model comparison using observable quantities.