  As an important problem in causal inference, we discuss the identification
and estimation of treatment effects (TEs) under limited overlap; that is, when
subjects with certain features belong to a single treatment group. We use a
latent variable to model a prognostic score which is widely used in
biostatistics and sufficient for TEs; i.e., we build a generative prognostic
model. We prove that the latent variable recovers a prognostic score, and the
model identifies individualized treatment effects. The model is then learned as
\beta-Intact-VAE--a new type of variational autoencoder (VAE). We derive the TE
error bounds that enable representations balanced for treatment groups
conditioned on individualized features. The proposed method is compared with
recent methods using (semi-)synthetic datasets.
