  Estimating the effect of a treatment on a given outcome, conditioned on a
vector of covariates, is central in many applications. However, learning the
impact of a treatment on a continuous temporal response, when the covariates
suffer extensively from measurement error and even the timing of the treatments
is uncertain, has not been addressed. We introduce a novel data-driven method
that can estimate treatment-response trajectories in this challenging scenario.
We model personalized treatment-response curves as a combination of parametric
response functions, hierarchically sharing information across individuals, and
a sparse Gaussian process for the baseline trend. Importantly, our model
considers measurement error not only in treatment covariates, but also in
treatment times, a problem which arises in practice for example when treatment
information is based on self-reporting. In a challenging and timely problem of
estimating the impact of diet on continuous blood glucose measurements, our
model leads to significant improvements in estimation accuracy and prediction.
