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

• URL: http://arxiv.org/abs/2601.03325v1
• Date: Tue, 06 Jan 2026 15:12:39 GMT
• Title: On the Identifiability of Regime-Switching Models with Multi-Lag Dependencies
• Abstract summary: We develop a general theoretical framework for multi-lag Regime-Switching Models (RSMs)<n>We prove identifiability of both the number of regimes and the multi-lag transitions in a nonlinear-Gaussian setting.<n>Our results hold in a fully unsupervised setting through architectural and noise assumptions.
• Score: 42.274922469778545
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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