Amortized Bayesian Multilevel Models (2408.13230v3)

Abstract: Multilevel models (MLMs) are a central building block of the Bayesian workflow. They enable joint, interpretable modeling of data across hierarchical levels and provide a fully probabilistic quantification of uncertainty. Despite their well-recognized advantages, MLMs pose significant computational challenges, often rendering their estimation and evaluation intractable within reasonable time constraints. Recent advances in simulation-based inference offer promising solutions for addressing complex probabilistic models using deep generative networks. However, the utility and reliability of deep learning methods for estimating Bayesian MLMs remains largely unexplored, especially when compared with gold-standard samplers. To this end, we explore a family of neural network architectures that leverage the probabilistic factorization of multilevel models to facilitate efficient neural network training and subsequent near-instant posterior inference on unseen datasets. We test our method on several real-world case studies and provide comprehensive comparisons to Stan's gold standard sampler, where possible. Finally, we provide an open-source implementation of our methods to stimulate further research in the nascent field of amortized Bayesian inference.

• The paper proposes a novel amortized inference method that dramatically accelerates posterior estimation in complex multilevel models.
• It utilizes hierarchical neural networks and probabilistic factorization to decompose joint posteriors, improving scalability and precision.
• Empirical validations on air passenger data, diffusion decision models, and handwriting style inference demonstrate robust and efficient performance.

An Expert Overview of "Amortized Bayesian Multilevel Models"

The paper "Amortized Bayesian Multilevel Models," authored by Daniel Habermann, Marvin Schmitt, Lars Kühmichel, Andreas Bulling, Stefan T. Radev, and Paul-Christian Bürkner, presents a detailed exploration and methodology for efficient Bayesian inference on multilevel models (MLMs) using amortized techniques. This work significantly addresses the computational challenges inherent in MLMs, especially the costly posterior estimation using traditional MCMC methods.

Introduction

Multilevel models are pivotal in modern Bayesian statistics due to their capability to model data hierarchically and provide comprehensive uncertainty quantification. Despite their advantages, the computational hurdles they present—particularly with large datasets and complex models—considerably limit their practical applicability. Standard MCMC techniques, despite improvements, remain infeasible for many practical problems due to their inherent computational demands. These challenges are exacerbated in scenarios requiring frequent model refitting, such as real-time data arrival or extensive Bayesian workflows involving cross-validation or simulation-based calibration.

Amortized Bayesian Inference

The authors propose leveraging recent advancements in neural density estimation to address these bottlenecks. By implementing amortized Bayesian inference (ABI), they aim to facilitate significantly faster posterior sampling after an initial, albeit substantial, training phase. Specifically, the paper details a method they term Multilevel Neural Posterior Estimation (ML-NPE), which involves the adaptation of neural density estimation techniques to hierarchical data structures.

Model Architecture

The core innovation hinges on the decomposition of the joint posterior into manageable components through hierarchical neural networks that parallel the probabilistic structure of MLMs. The architecture entails separate summary and inference networks at both the global and local levels, optimizing posterior approximations through specialized coupling layers and conditioning mechanisms.

Methodological Contributions

1. Hierarchical Networking: The paper outlines hierarchical network architectures that exploit the data's inherent structure, aiding efficient training and precise posterior inference.
2. Probabilistic Factorization: Building on exchangeability assumptions, the authors facilitate posterior factorization, subdividing inference tasks to align with the multilevel model's hierarchical nature.
3. Efficient and Scalable Inference: The method's implementation in the BayesFlow Python library ensures accessibility for further research and practical application, providing a scalable solution for Bayesian inference in MLMs.

Empirical Validation

The authors validate their method across three distinct case studies:

1. Air Passenger Traffic Analysis: The MLMs are used to model annual air passenger volumes between European countries and the US. The results are compared against Stan, demonstrating accurate posterior recovery and credible intervals, thereby showcasing the method's robustness in handling temporal dependencies and variable covariate spaces.
2. Diffusion Decision Model: The approach proves effective in cognitive science applications, modeling the decision-making process with varying subject-specific parameters. Leave-one-group-out cross-validation highlights the method's substantial computational advantages, enabling near-instant refitting.
3. Handwriting Style Inference: Leveraging a pre-trained generative network, the method's applicability to high-dimensional, unstructured data is demonstrated. The posterior inference scalability and accuracy underscore its potential in handling complex, data-intensive models.

Discussion and Future Directions

The successful empirical validation signifies the method's ability to extend Bayesian inference capabilities in significant scientific and practical scenarios. However, the authors identify future research directions, such as expanding the method to handle multi-level models beyond two hierarchical levels and improving training efficiency in low-data scenarios.

Conclusion

In summary, this paper provides a substantial methodological advancement in addressing the computational constraints of MLMs through amortized Bayesian inference. The integration of deep generative models and hierarchical architectures marks a notable stride in enhancing the scalability and efficiency of Bayesian approaches, with the potential to transform data-rich scientific inquiry.