Robust uncertainty quantification in popular estimators of the instantaneous reproduction number

The instantaneous reproduction number (${R}_t$) is a key measure of the rate of spread of an infectious disease. Correctly quantifying uncertainty in ${R}_t$ estimates is crucial for making well-informed decisions. Popular ${R}_t$ 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 real-time ${R}_t$ estimates. We derive novel model likelihoods for the smoothing parameters in both EpiEstim and EpiFilter and develop a Bayesian framework to automatically marginalize these parameters when fitting to epidemiological time-series data. This yields marginal posterior predictive distributions which prove integral to rigorous model evaluation. Applying our methods, we find that default parameterizations of these widely used estimators can negatively impact ${R}_t$ 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 ${R}_t$ inference, and facilitate model comparison using observable quantities.