📊 statistics

Identifiable Markov Switching Models with Instantaneous Effects and Exponential Families

This paper establishes the identifiability of latent regimes and regime-dependent causal structures in Markov Switching Models with instantaneous effects and exponential family noise, and proposes the FlowMSM framework to effectively detect these regimes and discover causal structures from non-stationary time series.

Original authors: Roel Hulsman, Carles Balsells-Rodas, Sara Magliacane

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Roel Hulsman, Carles Balsells-Rodas, Sara Magliacane

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the weather patterns of a city, but the city has a secret: it doesn't just have "sunny" or "rainy" days. Instead, it operates under different hidden "modes" or "regimes" that switch around. Sometimes the city is in "Monsoon Mode," sometimes "Drought Mode," and sometimes "Hurricane Mode."

The problem is, you can't see these modes directly. You only see the rain, the wind, and the temperature (the data). Furthermore, the rules for how the weather changes aren't the same in every mode. In "Monsoon Mode," a drop in pressure might instantly cause a flood. In "Drought Mode," that same drop might do nothing.

This is the challenge the paper tackles: How do we figure out which hidden mode the system is in, and what the specific rules are for that mode, just by looking at the messy data?

Here is a breakdown of their solution, using simple analogies.

1. The Problem: The "Chameleon" System

Most standard tools for analyzing time-series data (like stock prices or heart rates) assume the rules stay the same forever. But in the real world, systems change.

The "Instantaneous" Trap: Sometimes, changes happen so fast that our measurements miss the "in-between" steps. It's like taking a photo of a runner every 5 seconds. If they sprint from A to B in 1 second, your photos make it look like they teleported. The paper calls this an "instantaneous effect."
The "Non-Gaussian" Twist: Standard tools assume data follows a perfect "bell curve" (like average human height). But real data is often weird, lopsided, or has extreme outliers (like financial crashes or sudden glucose spikes). The authors' method works even when the data is "weird" (mathematically, from an "exponential family").

2. The Big Claim: "We Can Actually Solve This"

In math and science, "identifiability" means: Is there only one correct answer?
Imagine you have a locked box with a secret combination. If two different combinations produce the exact same lock sound, you can never know the real one. That's "unidentifiable."

The authors prove that their specific type of system is identifiable. They show that if you have enough data, there is only one unique way to explain the hidden modes and the rules governing them. They didn't just guess; they built a mathematical proof showing that the "bell curves" and "weird distributions" of the data are distinct enough that the hidden modes can't be confused with one another.

3. The Solution: FlowMSM (The "Smart Detective")

To find these hidden modes, they built a tool called FlowMSM. Think of it as a two-step detective agency:

Step 1: The Regime Detective (FlowMSM):
This part looks at the time-series data and asks, "Which hidden mode is active right now?"
- It uses a technique called Normalizing Flows. Imagine a piece of clay. You can stretch, twist, and squish it into complex shapes. This tool learns how to "squish" simple, predictable noise into the complex, weird shapes of your actual data. By tracking how the data gets "squished," it can tell which "mode" (which set of rules) is currently in charge.
- It groups the data into "windows" where the rules seem stable.
Step 2: The Causal Detective (The Partner):
Once the data is sorted into these stable windows, the tool hands the sorted chunks to a "Causal Discovery" partner.
- This partner asks: "In this specific mode, does A cause B, or does B cause A?"
- Because the data is now sorted by mode, the partner can find the true cause-and-effect rules without getting confused by the switching modes.

4. Why This is a Big Deal

Previous tools tried to do both steps at once or assumed the data was "nice" (Gaussian) and the modes didn't influence each other.

The Paper's Edge: Their method handles non-linear rules (where the effect isn't a straight line), instantaneous effects (fast changes), and weird data distributions.
The "Decoupling" Trick: Unlike other methods that try to guess the modes and the rules simultaneously (which can get stuck in a loop), FlowMSM separates the two jobs. First, it finds the modes. Then it finds the rules. This makes the whole process more stable and accurate.

5. Did It Work? (The Experiments)

The authors tested their detective on two types of cases:

Fake Data: They created computer simulations where they knew the "true" hidden modes and rules. FlowMSM successfully found the hidden modes and the rules, even when the data was noisy, non-linear, or had "instantaneous" jumps. It beat other popular methods, especially when the data wasn't a perfect bell curve.
Real Data (Finance): They applied it to the Fama-French Five-Factor Model (a famous way to explain stock returns) and Apple stock data.
- The Result: FlowMSM identified distinct "regimes" that lined up with real-world events. For example, it found a "Low Volatility" mode and a "High Volatility" mode that clearly captured periods like the 2008 financial crisis and the 2020 pandemic. Other methods failed to group these events together logically.

Summary

Think of the world as a chameleon changing colors. Old tools tried to guess the color by looking at a blurry photo and assuming the chameleon was always the same species.
This paper says: "No, the chameleon changes species too! But if we look closely at the texture of the skin (the math of exponential families) and how the colors shift instantly, we can prove exactly which species is present and how it behaves."

They built a tool (FlowMSM) that successfully separates the "species" (regimes) from the "behavior" (causal rules), even when the data is messy and the changes happen in a flash.

Technical Summary: Identifiable Markov Switching Models with Instantaneous Effects and Exponential Families

Problem Statement

Temporal systems in real-world environments, such as climate dynamics or glucose fluctuations in type-1 diabetes, frequently exhibit non-stationary behavior. A common modeling strategy represents this non-stationarity through discrete latent regimes, leading to Markov Switching Models (MSMs), a class of Hidden Markov Models (HMMs) with autoregressive dependencies. However, identifying these latent regimes and the underlying causal structures is challenging when:

Regime switches are frequent.
Dynamics are nonlinear and non-Gaussian.
Instantaneous effects exist between observed variables (e.g., due to coarse measurement granularity where causal effects occur faster than the sampling rate).

Existing identifiability results for finite-state HMMs do not trivially extend to MSMs due to autoregressive dependencies among observed variables. Furthermore, prior work on regime-switching models often assumes Gaussian settings, deterministic regime functions, or excludes instantaneous effects and causal discovery.

Methodology

Theoretical Framework: Identifiability

The authors establish the identifiability of latent regimes and regime-dependent causal structures under specific conditions. The data generating process is modeled as a regime-switching Structural Causal Model (SCM) where:

Regime Dynamics: Regimes follow a first-order stationary Markov chain (dependencies between regimes are allowed).
Causal Mechanisms: Within each regime, the system follows a structural equation with nonlinear lagged and instantaneous effects.
Noise: Exogenous noise belongs to a continuous minimal regular exponential family (e.g., Gaussian, Gamma, Laplace).

The paper proves that the induced MSM is identifiable up to permutation (i.e., the data likelihood uniquely determines the latent regimes and mixture components, up to a relabeling of regimes) under the following assumptions:

Exponential Family Noise: The sufficient statistic of the noise is real-analytic and has a rich image (contains an open set).
Functional Restrictions: The mapping functions are unique per regime, act as pointwise diffeomorphisms, and are jointly real-analytic.
Noise Automorphisms: The noise distribution has a trivial automorphism class, or the mappings are restricted via monotone canonicalisation (e.g., no rotations).
Sufficient Variability: The regime-dependent effects can be captured by a finite-order polynomial factorization of the sufficient statistic, ensuring distinct regimes map to distinct natural parameters.

This theory subsumes non-temporal mixtures of causal models and extends previous Gaussian-only results to broader exponential families and instantaneous effects.

Algorithm: FlowMSM

To operationalize this theory, the authors introduce FlowMSM, a regime detection framework that decouples regime detection from causal discovery.

Regime Detection: FlowMSM employs a Generalised Expectation-Maximisation (GEM) algorithm. It models the initial and transition distributions using Conditional Normalising Flows (CNFs), specifically Neural Spline Flows (NSF). This allows the model to capture complex, non-Gaussian, and nonlinear densities. The flow parameters are shared across regimes but conditioned on regime-specific embeddings and lagged variables.
Causal Discovery: Once regimes are identified (via Maximum a Posteriori assignment), the time series is split into clusters corresponding to each regime. Standard stationary causal discovery methods (e.g., VARLiNGAM, DYNOTEARS, Rhino) are then applied independently to each cluster to recover the regime-dependent window graphs.

Key Contributions

Identifiability Theory: The paper provides sufficient conditions for the identifiability of regime-switching SCMs with dependencies between regimes, nonlinear lagged/instantaneous effects, and exponential family noise. This generalizes existing theory beyond Gaussian assumptions.
FlowMSM Framework: A novel estimation method combining conditional normalising flows with GEM for robust regime detection in non-Gaussian, nonlinear settings.
Decoupled Approach: Unlike prior methods that iterate between regime detection and causal discovery within restrictive model classes (like Location Scale Noise Models), FlowMSM decouples these phases, allowing for a broader class of causal models.
Empirical Validation: Extensive experiments on synthetic benchmarks and a financial economics dataset (Fama-French five-factor model) demonstrate the method's effectiveness.

Experimental Results

Synthetic Benchmarks

Experiments were conducted on synthetic datasets with $N=100$ time series, $T=100$ time steps, $D=10$ dimensions, and $K=3$ regimes. The datasets included:

Linear SVAR with Gaussian and Exponential Family (Gamma) noise.
Nonlinear Additive Noise Models (ANM) with Exponential Family noise.
Location Scale Noise Models (LSNM) with Exponential Family noise (the most challenging setting).

Findings:

Regime Detection: FlowMSM achieved high Normalised Mutual Information (NMI) and Adjusted Rand Index (ARI) scores across all settings, significantly outperforming baselines (iMSM, R-PCMCI, CASTOR, FANTOM) in nonlinear and non-Gaussian scenarios. Baselines relying on Gaussian assumptions degraded substantially in non-Gaussian settings.
Causal Discovery: When paired with appropriate discovery methods (e.g., VARLiNGAM for non-Gaussian, Rhino for LSNM), FlowMSM recovered causal structures with high accuracy and low Structural Hamming Distance (SHD). Baselines generally failed to achieve competitive performance, likely due to difficulties in handling frequent regime switches and non-Gaussian dynamics.
Robustness: FlowMSM remained effective even when the number of regimes or lags was overspecified.

Real-World Application: Fama-French Model

The authors applied FlowMSM to daily data from the Fama-French five-factor model and Apple (AAPL) stock returns.

Regime Detection: FlowMSM identified two dominant regimes aligned with periods of low and high market volatility (including the 2008 crisis and COVID-19 pandemic), which correlated with the VIX index. Other methods produced regimes that did not map clearly to market events.
Causal Interpretation: The estimated causal graphs suggested that during stable markets (Regime 4), causal dependencies were absent (consistent with the Efficient Market Hypothesis), while during market distress (Regime 5), causal connections emerged. However, the authors note that causal interpretations of the Fama-French factors driving AAPL returns were limited and not fully supported by the data.

Significance and Claims

The paper claims to provide a principled framework for regime identification in non-stationary time series.

Theoretical Rigor: By establishing identifiability under mild assumptions (real-analyticity, exponential family noise), the work justifies the recovery of latent structures that were previously considered unidentifiable in the presence of instantaneous effects and regime dependencies.
Practical Utility: The FlowMSM framework offers a practical solution for detecting latent regimes and discovering causal structures in complex, non-Gaussian environments where traditional Gaussian MSMs fail.
Modesty: The authors acknowledge limitations, noting that the assumption of regime independence from observations ( $R_t \not\to X_{0:T}$ ) is strong and that the theory relies on real-analytic diffeomorphisms. They also caution that causal interpretations in real-world financial data may be limited in scope and that monthly data (with fewer samples) failed to yield meaningful results, highlighting the data requirements of the approach.

The work does not claim to solve all non-stationary causal discovery problems but offers a significant step forward by relaxing Gaussian assumptions and explicitly handling instantaneous effects within a theoretically grounded identifiability framework.