On the Identifiability of Regime-Switching Models with Multi-Lag Dependencies

TL;DR

This paper develops a theoretical framework for the identifiability of multi-lag regime-switching models, ensuring unique parameter recovery and interpretability in complex time series across various scientific domains.

Contribution

It provides the first comprehensive identifiability conditions for deep regime-switching models, including Markov switching and switching dynamical systems, with practical neural network-based estimators.

Findings

Proves identifiability of regimes and transitions in Markov switching models.

Establishes conditions for latent variable and causal graph identifiability in switching dynamical systems.

Validates the theoretical results on synthetic and real-world datasets from neuroscience, finance, and climate.

Abstract

Identifiability is central to the interpretability of deep latent variable models, ensuring parameterisations are uniquely determined by the data-generating distribution. However, it remains underexplored for deep regime-switching time series. We develop a general theoretical framework for multi-lag Regime-Switching Models (RSMs), encompassing Markov Switching Models (MSMs) and Switching Dynamical Systems (SDSs). For MSMs, we formulate the model as a temporally structured finite mixture and prove identifiability of both the number of regimes and the multi-lag transitions in a nonlinear-Gaussian setting. For SDSs, we establish identifiability of the latent variables up to permutation and scaling via temporal structure, which in turn yields conditions for identifiability of regime-dependent latent causal graphs (up to regime/node permutations). Our results hold in a fully unsupervised setting through architectural and noise assumptions that are directly enforceable via neural network design. We complement the theory with a flexible variational estimator that satisfies the assumptions and validate the results on synthetic benchmarks. Across real-world datasets from neuroscience, finance, and climate, identifiability leads to more trustworthy interpretability analysis, which is crucial for scientific discovery.