smoothbp: Fast Bayesian Hierarchical Piecewise Regression with Smoothed Transitions and Spike-and-Slab Model Selection

The detection of structural changes in data over time or space has long been a core challenge in statistics. Early methods like classical segmented regression assumed abrupt shifts, but real-world phenomena often exhibit gradual transitions. Recent advances include Bayesian approaches such as mcp and brms, which support multiple change-points and hierarchical structures. However, these often assume instantaneous changes and lack integrated support for smooth transitions or automatic model selection. The need for models that can handle multiple, gradual shifts with hierarchical effects has driven research into more flexible frameworks. smoothbp builds upon these developments, combining logistic smoothing with hierarchical Bayesian modeling, leveraging efficient computational techniques to address existing limitations.

Existing models either assume sharp, instantaneous change-points or lack the ability to handle hierarchical data structures with multiple, gradually transitioning shifts. This creates a gap in accurately modeling phenomena like climate regimes, disease progression, or financial market shifts, where changes are often smooth and heterogeneous across groups. Additionally, current software tools are limited in supporting automatic selection of multiple change-points within a hierarchical context, often requiring manual intervention or simplified assumptions. The core challenge is to develop a scalable, flexible Bayesian framework that can accurately detect multiple, smooth change-points, incorporate hierarchical random effects, and perform automatic model selection efficiently.

smoothbp introduces several key innovations: 1) Logistic-based smooth transition functions that model gradual shifts; 2) Hierarchical Bayesian structure allowing random effects on change-point locations and parameters; 3) Rust-implemented sampler combining conjugate Gibbs updates with HMC for high efficiency; 4) Spike-and-Slab priors enabling automatic selection of active change-points. These innovations collectively enable a realistic, flexible modeling of complex data with multiple, evolving structural changes, overcoming the rigidity of traditional models. The integration of high-performance computing with advanced Bayesian techniques sets this work apart from prior approaches.

• �� Define a hierarchical Bayesian model with multiple change-points, where each segment’s response is modeled as a linear function with parameters (intercept, slope) and change-points characterized by location (ωk) and sharpness (ρk).
• �� Use a logistic function to smoothly transition between segments, with parameters allowing the transition width to be estimated from data.
• �� Incorporate hierarchical random effects on change-point locations (ωk,j) to account for group heterogeneity, with optional non-centered parametrization to improve mixing.
• �� Apply Spike-and-Slab priors on slope changes (δk), with binary indicators (γk) controlling inclusion, facilitating automatic selection of relevant change-points.
• �� Implement a hybrid sampler: conjugate Gibbs updates for linear parameters, HMC for nonlinear parameters, with Rust ensuring computational efficiency.
• �� Support fixed parameters, boundary constraints, and auxiliary functions for prior specification, model diagnostics, and simulation.
• �� Validate performance through extensive simulation, calibration, and real data applications, ensuring robustness and scalability.

The experimental setup involves simulation studies with varying numbers of change-points, hierarchical structures, and smoothness levels. Metrics include bias, coverage probability, and effective sample size. Comparisons against brms and mcp focus on sampling efficiency, accuracy in change-point detection, and parameter recovery. Real datasets from ecological spatial-temporal studies and epidemiological time series are analyzed to demonstrate practical utility. Hyperparameters such as prior variances and transition sharpness are tuned based on prior predictive checks. Ablation studies assess the impact of hierarchical effects and smoothing parameters on inference quality. The experiments confirm the model’s robustness, efficiency, and applicability across diverse scenarios.

Results show that smoothbp accurately recovers true change-point locations with mean bias below 0.1 time units, and posterior intervals achieve 95% coverage in over 90% of simulations. Its sampling efficiency surpasses that of brms and mcp by 20-30%, especially in hierarchical multi-breakpoint models. The spike-and-slab prior correctly identifies active change-points with posterior inclusion probabilities above 0.8 in most cases. In real data applications, it captures gradual climate regime shifts and heterogeneous intervention effects, outperforming traditional abrupt-change models. Calibration studies confirm the reliability of posterior estimates and model selection procedures.

smoothbp is ideal for analyzing ecological data with gradual climate shifts, epidemiological studies tracking disease progression, and financial data exhibiting structural market changes. Its ability to incorporate hierarchical effects and perform automatic change-point selection makes it suitable for large-scale, complex datasets. Practitioners can use it to detect subtle, evolving patterns, inform policy interventions, or forecast future trends with quantified uncertainty. Its flexible framework supports diverse covariates and model specifications, broadening its applicability across scientific disciplines.

The model may face challenges with extremely noisy data or when change-points are highly ambiguous, leading to potential misidentification. Computational costs increase with model complexity, especially in high-dimensional settings with many change-points and random effects. Sensitivity to prior choices requires careful tuning, and the current implementation may struggle with real-time analysis or very large datasets. Future work should focus on algorithmic optimization, extending to non-linear and non-parametric models, and developing more automated prior elicitation methods.