Causal structure learning from time series data is a major scientific challenge. Extant algorithms assume that measurements occur sufficiently quickly; more precisely, they assume approximately equal system and measurement timescales. In many domains, however, measurements occur at a significantly slower rate than the underlying system changes, but the size of the timescale mismatch is often unknown. This paper develops three causal structure learning algorithms, each of which discovers all dynamic causal graphs that explain the observed measurement data, perhaps given undersampling. That is, these algorithms all learn causal structure in a "rate-agnostic" manner: they do not assume any particular relation between the measurement and system timescales. We apply these algorithms to data from simulations to gain insight into the challenge of undersampling.

Variational methods that rely on a recognition network to approximate the posterior of directed graphical models offer better inference and learning than previous methods. Recent advances that exploit the capacity and flexibility in this approach have expanded what kinds of models can be trained. However, as a proposal for the posterior, the capacity of the recognition network is limited, which can constrain the representational power of the generative model and increase the variance of Monte Carlo estimates. To address these issues, we introduce an iterative refinement procedure for improving the approximate posterior of the recognition network and show that training with the refined posterior is competitive with state-of-the-art methods. The advantages of refinement are further evident in an increased effective sample size, which implies a lower variance of gradient estimates.