  It is of particular interests in many application fields to draw doubly
robust inference of a logistic partially linear model with the predictor
specified as combination of a targeted low dimensional linear parametric
function and a nuisance nonparametric function. In recent, Tan (2019) proposed
a simple and flexible doubly robust estimator for this purpose. They introduced
the two nuisance models, i.e. nonparametric component in the logistic model and
conditional mean of the exposure covariates given the other covariates and
fixed response, and specified them as fixed dimensional parametric models.
Their framework could be potentially extended to machine learning or high
dimensional nuisance modelling exploited recently, e.g. in Chernozhukovet al.
(2018a,b) and Smucler et al. (2019); Tan (2020). Motivated by this, we derive
the debiased/double machine learning logistic partially linear model in this
note. For construction of the nuisance models, we separately consider the use
of high dimensional sparse parametric models and general machine learning
methods. By deriving certain moment equations to calibrate the first order bias
of the nuisance models, we preserve a model double robustness property on high
dimensional ultra-sparse nuisance models. We also discuss and compare the
underlying assumption of our method with debiased LASSO (Van deGeer et al.,
2014). To implement the machine learning proposal, we design a full model
refitting procedure that allows the use of any blackbox conditional mean
estimation method in our framework. Under the machine learning setting, our
method is rate doubly robust in a similar sense as Chernozhukov et al. (2018a).
