Identifiable Markov Switching Models with Instantaneous Effects and Exponential Families

Temporal systems often exhibit non-stationary behaviour, such as seasonal climate variation or glucose fluctuations in patients with type-1 diabetes. One way to model non-stationarity is through discrete latent *regimes*, *i.e.*, stationary segments of time. Such systems induce a *Markov Switching Model* (MSM), a class of Hidden Markov Models with autoregressive dependencies among latent regimes and observed variables. Identifying latent regimes is challenging in the presence of frequent regime switches and nonlinear and non-Gaussian dynamics, particularly when there are *instantaneous effects* between the variables, *e.g.*, due to slow rates of measurements. In this work, we establish the identifiability of both latent regimes and regime-dependent causal structures under temporal regime dependencies, nonlinear lagged and instantaneous effects, and independent noise from the exponential family. Our identifiability theory subsumes non-temporal mixtures of causal models. Furthermore, we introduce $\texttt{FlowMSM}$, a regime detection framework that can be paired with any stationary causal discovery method to recover regime-dependent causal structures. Experiments on synthetic benchmarks and a financial economics dataset demonstrate the effectiveness of our approach to detect latent regimes and discover causal structures from non-stationary time series.

Many real-world systems, like weather patterns or glucose fluctuations in diabetes patients, tend to shift their behaviour over time, moving between hidden phases called *regimes*. Detecting such regimes in time series is difficult because the underlying causal dynamics keep changing and could be highly complex, which particularly occurs when causal effects appear faster than the rate at which the data is measured, called *instantaneous effects*. We prove that it is possible to uncover both the hidden regimes and the causal structure in a broad class of causal models with exponential family noise. Building on this theory, we develop a framework called $\texttt{FlowMSM}$ that automatically detects the hidden regimes and can be extended to discover a set of regime-dependent temporal causal structures. We apply our work to synthetic data and a financial economics dataset, showcasing that we can effectively detect regimes and discover causal structures in complex non-stationary environments. For example, based on stock market indicators, our method differentiates stable from volatile periods, such as the 2008 financial crisis, and explores causal interpretations of the efficient market hypothesis.