  In epidemiological modelling, the instantaneous reproduction number, $R_t$,
is important to understand the transmission dynamics of infectious diseases.
Current $R_t$ estimates often suffer from problems such as lagging, averaging
and uncertainties demoting the usefulness of $R_t$. To address these problems,
we propose a new method in the framework of sequential Bayesian inference where
a Data Assimilation approach is taken for $R_t$ estimation, resulting in the
state-of-the-art 'DAR$_t$' system for $R_t$ estimation. With DAR$_t$, the
problem of time misalignment caused by lagging observations is tackled by
incorporating observation delays into the joint inference of infections and
$R_t$; the drawback of averaging is improved by instantaneous updating upon new
observations and a model selection mechanism capturing abrupt changes caused by
interventions; the uncertainty is quantified and reduced by employing Bayesian
smoothing. We validate the performance of DAR$_t$ through simulations and
demonstrate its power in revealing the transmission dynamics of COVID-19.
