  In some causal inference scenarios, the treatment variable is measured
inaccurately, for instance in epidemiology or econometrics. Failure to correct
for the effect of this measurement error can lead to biased causal effect
estimates. Previous research has not studied methods that address this issue
from a causal viewpoint while allowing for complex nonlinear dependencies and
without assuming access to side information. For such a scenario, this study
proposes a model that assumes a continuous treatment variable that is
inaccurately measured. Building on existing results for measurement error
models, we prove that our model's causal effect estimates are identifiable,
even without knowledge of the measurement error variance or other side
information. Our method relies on a deep latent variable model in which
Gaussian conditionals are parameterized by neural networks, and we develop an
amortized importance-weighted variational objective for training the model.
Empirical results demonstrate the method's good performance with unknown
measurement error. More broadly, our work extends the range of applications in
which reliable causal inference can be conducted.
