  An essential and challenging problem in causal inference is causal effect
estimation from observational data. The problem becomes more difficult with the
presence of unobserved confounding variables. The front-door adjustment is a
practical approach for dealing with unobserved confounding variables. However,
the restriction for the standard front-door adjustment is difficult to satisfy
in practice. In this paper, we relax some of the restrictions by proposing the
concept of conditional front-door (CFD) adjustment and develop the theorem that
guarantees the causal effect identifiability of CFD adjustment. Furthermore, as
it is often impossible for a CFD variable to be given in practice, it is
desirable to learn it from data. By leveraging the ability of deep generative
models, we propose CFDiVAE to learn the representation of the CFD adjustment
variable directly from data with the identifiable Variational AutoEncoder and
formally prove the model identifiability. Extensive experiments on synthetic
datasets validate the effectiveness of CFDiVAE and its superiority over
existing methods. The experiments also show that the performance of CFDiVAE is
less sensitive to the causal strength of unobserved confounding variables. We
further apply CFDiVAE to a real-world dataset to demonstrate its potential
application.
