  Treatment effect estimation in continuous time is crucial for personalized
medicine. However, existing methods for this task are limited to point
estimates of the potential outcomes, whereas uncertainty estimates have been
ignored. Needless to say, uncertainty quantification is crucial for reliable
decision-making in medical applications. To fill this gap, we propose a novel
Bayesian neural controlled differential equation (BNCDE) for treatment effect
estimation in continuous time. In our BNCDE, the time dimension is modeled
through a coupled system of neural controlled differential equations and neural
stochastic differential equations, where the neural stochastic differential
equations allow for tractable variational Bayesian inference. Thereby, for an
assigned sequence of treatments, our BNCDE provides meaningful posterior
predictive distributions of the potential outcomes. To the best of our
knowledge, ours is the first tailored neural method to provide uncertainty
estimates of treatment effects in continuous time. As such, our method is of
direct practical value for promoting reliable decision-making in medicine.
