  We investigate the relationship between system identification and
intervention design in dynamical systems. While previous research demonstrated
how identifiable representation learning methods, such as Independent Component
Analysis (ICA), can reveal cause-effect relationships, it relied on a passive
perspective without considering how to collect data. Our work shows that in
Gaussian Linear Time-Invariant (LTI) systems, the system parameters can be
identified by introducing diverse intervention signals in a multi-environment
setting. By harnessing appropriate diversity assumptions motivated by the ICA
literature, our findings connect experiment design and representational
identifiability in dynamical systems. We corroborate our findings on synthetic
and (simulated) physical data. Additionally, we show that Hidden Markov Models,
in general, and (Gaussian) LTI systems, in particular, fulfil a generalization
of the Causal de Finetti theorem with continuous parameters.

Machine Learning;Computational Engineering, Finance, and Science;Systems and Control;Systems and Control;Methodology