Non-stationary time series are common in many domains and may exhibit evolving causal relationships over time. The CaReBayes framework combines regime detection, temporal causal discovery, regime-specific Bayesian structural autoregressive modeling, regime classification, and regime-conditioned forecasting with posterior predictive uncertainty. CaReBayes segments time series into regimes by identifying changes in estimated temporal causal structure, fits Bayesian models per regime, assigns new observations to regimes based on predictive compatibility with learned Bayesian structures, and issues regime-specific forecasts. According to the authors, CaReBayes outperformed benchmark linear models on synthetic datasets and Ontario COVID-19 time series while producing interpretable regime-dependent causal graphs.
Forecasting and interpretability are central goals when analyzing time-indexed measurements. These goals are complicated when a system is non-stationary, meaning its statistical properties change over time. Common non-stationarity takes the form of trends and seasonality, but an important type is causal non-stationarity, where the underlying causal structure—represented as a directed acyclic graph (DAG) in structural causal models—evolves over time.
When the causal graph is stable, the system is causally stationary. Conversely, when causal relationships change, the system can be described as regime-dependent: contiguous intervals (regimes) during which the joint distribution and causal structure remain consistent but differ between intervals. Regime dependence is common in finance, epidemiology, and engineering, often driven by policy changes, seasonality, or other interventions.
Many standard forecasting methods assume stationarity or model changes only at the level of distributions, and so they may miss shifts in causal relationships. Prior methods addressing regime dynamics include Markov-switching VAR, structural time series, threshold and hysteretic autoregressive models, switching dynamical systems, and state-space approaches. Separately, causal discovery methods have been adapted to non-stationary data to estimate causal graphs across time or regimes. However, a gap exists between methods that model regime-dependent statistical dynamics and those that explicitly identify changing causal structure while producing regime-conditioned forecasts with uncertainty quantification.
To evaluate the proposed framework, the authors compared CaReBayes against two linear forecasting baselines: the Vector Autoregressive (VAR) model and the Markov-Switching VAR (MS-VAR) model. Synthetic datasets in this study were generated from linear processes; therefore, linear baselines were used for fairness and interpretability.
This work contrasted CaReBayes with two widely used, linear benchmarks.
The VAR model represents each variable as a linear combination of its own and other variables' past values. It assumes a stable data-generating process and was estimated via maximum likelihood under Gaussian noise assumptions. VAR serves as a regime-agnostic baseline that does not model structural changes.
The MS-VAR extends VAR by allowing parameters to vary across a finite number of latent regimes governed by a first-order Markov process. Each regime has its own intercepts, autoregressive coefficients, and noise structure. Parameters and regime probabilities are estimated via an EM algorithm with Hamilton filtering for forward–backward inference. MS-VAR captures regime-dependent dynamics statistically but does not estimate regime-specific causal graphs or provide causal interpretation in the SCM sense.
CaReBayes is a four-stage framework that integrates causal discovery and Bayesian forecasting to handle regime-dependent non-stationarity. The stages are: (1) regime segmentation with causal discovery, (2) regime-specific Bayesian structural autoregressive modeling, (3) regime classification, and (4) forecasting with uncertainty quantification.
Regimes are identified by detecting changes in the estimated temporal causal structure rather than by pre-specified threshold variables. The framework uses temporal causal discovery methods applied across candidate segments to infer regime-dependent causal graphs. A grid-search procedure is introduced to automate regime-dependent causal discovery, searching over segmentation configurations and selecting partitions based on predictive performance and structural consistency. Specific implementation details of the causal discovery algorithm and grid-search hyperparameters were provided in the full article.
For each discovered regime, CaReBayes fits a Bayesian structural autoregressive model that respects the regime-specific causal graph. This produces parameter estimates and posterior distributions conditional on the regime's structure, enabling posterior predictive inference that accounts for uncertainty in both parameters and noise.
To forecast, CaReBayes must assign future observations to one of the learned regimes. The framework classifies new observations into regimes based on predictive compatibility with the regime-specific Bayesian structural models. The classifier leverages the learned Bayesian structures to determine which regime's predictive distribution best explains the newly observed covariates and lags.
Once a regime is assigned, CaReBayes issues regime-conditioned forecasts using the corresponding Bayesian structural autoregressive model. Forecasts include posterior predictive uncertainty, reflecting parameter and observation-level uncertainty under the selected regime model.
The authors evaluated CaReBayes on synthetic linear datasets and on Ontario COVID-19 time series. Baselines were VAR and MS-VAR. The abstract reports that CaReBayes outperformed these benchmarks in forecasting performance and additionally produced regime-dependent causal graphs to aid interpretability. Full numerical results, performance metrics, and experimental settings are reported in the article's Results and Figures sections.
CaReBayes aims to bridge a methodological gap by combining discrete regime-dependent temporal causal discovery, predictive-error-based classification of future observations into learned causal regimes, and regime-conditioned Bayesian structural forecasting. Unlike threshold or hysteretic models, CaReBayes does not require prior specification of an observed transition variable or threshold boundaries; regimes are inferred from changes in temporal causal structure.
The framework is presented as the first to jointly perform these components and to produce regime-specific causal summaries alongside probabilistic forecasts. The authors emphasize interpretability via regime-dependent causal graphs and improved forecasting accuracy on their evaluated datasets.
The article outlines directions for future development, details of which are provided in the full text. Any additional experiments, scalability assessments, or extensions to non-linear generative processes beyond the linear-synthetic and COVID-19 case studies are discussed by the authors in the Future work section.
CaReBayes integrates regime detection, causal discovery, regime classification, and Bayesian forecasting to address regime-dependent causal non-stationarity. It identifies regimes from changes in temporal causal structure, fits Bayesian structural autoregressive models per regime, classifies new observations by predictive compatibility, and issues regime-conditioned forecasts with posterior predictive uncertainty. The framework reportedly outperformed linear VAR and MS-VAR benchmarks on the study datasets and provides regime-specific causal graphs to support interpretability.
The authors provided accessible data and code at https://github.com/bdmossop/CaReBayes/tree/main. For detailed algorithms, hyperparameters, and numerical results, consult the full article.