  We propose causal recurrent variational autoencoder (CR-VAE), a novel
generative model that is able to learn a Granger causal graph from a
multivariate time series x and incorporates the underlying causal mechanism
into its data generation process. Distinct to the classical recurrent VAEs, our
CR-VAE uses a multi-head decoder, in which the $p$-th head is responsible for
generating the $p$-th dimension of $\mathbf{x}$ (i.e., $\mathbf{x}^p$). By
imposing a sparsity-inducing penalty on the weights (of the decoder) and
encouraging specific sets of weights to be zero, our CR-VAE learns a sparse
adjacency matrix that encodes causal relations between all pairs of variables.
Thanks to this causal matrix, our decoder strictly obeys the underlying
principles of Granger causality, thereby making the data generating process
transparent. We develop a two-stage approach to train the overall objective.
Empirically, we evaluate the behavior of our model in synthetic data and two
real-world human brain datasets involving, respectively, the
electroencephalography (EEG) signals and the functional magnetic resonance
imaging (fMRI) data. Our model consistently outperforms state-of-the-art time
series generative models both qualitatively and quantitatively. Moreover, it
also discovers a faithful causal graph with similar or improved accuracy over
existing Granger causality-based causal inference methods. Code of CR-VAE is
publicly available at https://github.com/hongmingli1995/CR-VAE.
