  Collaboration between different data centers is often challenged by
heterogeneity across sites. To account for the heterogeneity, the
state-of-the-art method is to re-weight the covariate distributions in each
site to match the distribution of the target population. Nevertheless, this
method could easily fail when a certain site couldn't cover the entire
population. Moreover, it still relies on the concept of traditional
meta-analysis after adjusting for the distribution shift.
  In this work, we propose a collaborative inverse propensity score weighting
estimator for causal inference with heterogeneous data. Instead of adjusting
the distribution shift separately, we use weighted propensity score models to
collaboratively adjust for the distribution shift. Our method shows significant
improvements over the methods based on meta-analysis when heterogeneity
increases. To account for the vulnerable density estimation, we further discuss
the double machine method and show the possibility of using nonparametric
density estimation with d<8 and a flexible machine learning method to guarantee
asymptotic normality. We propose a federated learning algorithm to
collaboratively train the outcome model while preserving privacy. Using
synthetic and real datasets, we demonstrate the advantages of our method.
