Hamiltonian Monte Carlo for Hierarchical Models

Hierarchical models have become fundamental in statistical science for modeling multi-level dependencies, such as in spatial, temporal, and multilevel data. Early methods like Gibbs sampling and Metropolis-Hastings provided practical tools but faced limitations with high-dimensional, correlated posteriors. The advent of Hamiltonian Monte Carlo (HMC) introduced gradient-based exploration, significantly improving efficiency. Recent developments incorporated geometric information, notably Riemannian HMC, to adapt to local curvature. These advances address the geometric complexity of models like funnel distributions, but practical implementation remains challenging due to computational costs and metric selection. The paper builds on these foundations, aiming to make geometric HMC more robust and scalable for hierarchical models.

The core challenge in hierarchical Bayesian inference is the 'funnel' distribution, which exhibits regions of high curvature and strong correlation, impeding efficient exploration. Standard algorithms like Gibbs and Metropolis become trapped or explore slowly in these regions, leading to poor mixing and convergence issues. Existing parameterizations (centered vs. non-centered) offer partial solutions but are sensitive to data sparsity and model structure. The geometric complexity of the posterior demands methods that can adapt locally to the curvature, yet such methods are computationally intensive and difficult to implement reliably. Overcoming these obstacles is crucial for scalable inference in complex models.

The paper's primary innovation is embedding Riemannian geometry into HMC via the SoftAbs transformation, which regularizes the Hessian of the target distribution to produce a positive-definite metric that adapts locally to curvature. This allows the sampler to navigate highly correlated, complex geometries efficiently. The integration of automatic differentiation streamlines gradient computations, while the No-U-Turn sampler dynamically adjusts trajectory lengths, maintaining detailed balance. This combination results in a scalable, geometry-aware HMC that significantly outperforms traditional methods in hierarchical models, especially in 'funnel'-shaped posteriors.

• �� Construct Hamiltonian system combining target distribution and auxiliary momentum variables. • Derive Riemannian metric using SoftAbs transformation of the Hessian, ensuring positive definiteness. • Compute gradients via automatic differentiation for precise, efficient updates. • Implement leapfrog integrator respecting the geometry, preserving symplectic structure. • Use No-U-Turn criterion to adaptively determine trajectory length, avoiding retracing. • Tune step size and metric parameters through preliminary warm-up phases, balancing exploration and stability. • Apply the method to high-dimensional hierarchical models, comparing with baseline algorithms, and validate improvements in ESS and convergence speed.

Experiments involved simulated hierarchical normal models with 800 latent parameters, comparing centered and non-centered parameterizations. Metrics included effective sample size (ESS), sampling time, and convergence diagnostics. Real data applications involved multi-level random effects models with complex correlations. The hyperparameters, such as step size and metric regularization parameters, were tuned via adaptive schemes. The performance was benchmarked against Gibbs and Metropolis, demonstrating superior efficiency and stability. Additional ablation studies assessed the impact of geometric adaptation and automatic differentiation, confirming their critical roles in performance gains.

In high-dimensional simulations, the non-centered parameterization with Riemannian HMC achieved over tenfold increase in ESS and reduced sampling time by a factor of 20 compared to traditional algorithms. Real-world models showed faster convergence, more thorough posterior exploration, and robustness in highly correlated regions. The geometric approach effectively mitigated the funnel's curvature effects, enabling exploration of parameter spaces that were previously intractable. These results establish the method as a practical and scalable solution for complex hierarchical Bayesian inference.

The method is applicable to a broad class of hierarchical Bayesian models, including spatial-temporal models, deep Bayesian neural networks, and multi-level social science models. It enables efficient inference in high-dimensional, strongly correlated posteriors, facilitating scientific discovery and industrial applications requiring precise uncertainty quantification. The approach can be integrated into existing probabilistic programming frameworks like Stan, enhancing their capability to handle complex models with minimal manual tuning.

Despite its advantages, the approach incurs significant computational costs due to Hessian calculations and metric regularization, limiting scalability in extremely high-dimensional models. The choice of regularization parameters and metric form requires expert tuning, which can be challenging. In models with non-smooth or non-differentiable posteriors, the Hessian-based metric may be ill-conditioned, affecting stability. Future work should focus on automating metric learning, reducing computational overhead, and extending applicability to non-smooth models.