  Methods utilizing instrumental variables have been a fundamental statistical
approach to estimation in the presence of unmeasured confounding, usually
occurring in non-randomized observational data common to fields such as
economics and public health. However, such methods usually make constricting
linearity and additivity assumptions that are inapplicable to the complex
modeling challenges of today. The growing body of observational data being
collected will necessitate flexible regression modeling while also being able
to control for confounding using instrumental variables. Therefore, this
article presents a nonlinear instrumental variable regression model based on
Bayesian regression tree ensembles to estimate such relationships, including
interactions, in the presence of confounding. One exciting application of this
method is to use genetic variants as instruments, known as Mendelian
randomization. Body mass index is one factor that is hypothesized to have a
nonlinear relationship with cardiovascular risk factors such as blood pressure
while interacting with age. Heterogeneity in patient characteristics such as
age could be clinically interesting from a precision medicine perspective where
individualized treatment is emphasized. We present our flexible Bayesian
instrumental variable regression tree method with an example from the UK
Biobank where body mass index is related to blood pressure using genetic
variants as the instruments.
