  Granger causality analysis, as one of the most popular time series causality
methods, has been widely used in the economics, neuroscience. However,
unobserved confounders is a fundamental problem in the observational studies,
which is still not solved for the non-linear Granger causality. The application
works often deal with this problem in virtue of the proxy variables, who can be
treated as a measure of the confounder with noise. But the proxy variables has
been proved to be unreliable, because of the bias it may induce. In this paper,
we try to "recover" the unobserved confounders for the Granger causality. We
use a generative model with latent variable to build the relationship between
the unobserved confounders and the observed variables(tested variable and the
proxy variables). The posterior distribution of the latent variable is adopted
to represent the confounders distribution, which can be sampled to get the
estimated confounders. We adopt the variational autoencoder to estimate the
intractable posterior distribution. The recurrent neural network is applied to
build the temporal relationship in the data. We evaluate our method in the
synthetic and semi-synthetic dataset. The result shows our estimated
confounders has a better performance than the proxy variables in the non-linear
Granger causality with multiple proxies in the semi-synthetic dataset. But the
performances of the synthetic dataset and the different noise level of proxy
seem terrible. Any advice can really help.
