Hamiltonian Monte Carlo Inference of Marginalized Linear Mixed-Effects Models (2410.24079v3)

Abstract: Bayesian reasoning in linear mixed-effects models (LMMs) is challenging and often requires advanced sampling techniques like Markov chain Monte Carlo (MCMC). A common approach is to write the model in a probabilistic programming language and then sample via Hamiltonian Monte Carlo (HMC). However, there are many ways a user can transform a model that make inference more or less efficient. In particular, marginalizing some variables can greatly improve inference but is difficult for users to do manually. We develop an algorithm to easily marginalize random effects in LMMs. A naive approach introduces cubic time operations within an inference algorithm like HMC, but we reduce the running time to linear using fast linear algebra techniques. We show that marginalization is always beneficial when applicable and highlight improvements in various models, especially ones from cognitive sciences.

• The paper introduces an algorithm that analytically marginalizes random effects in LMMs, reducing computational complexity from cubic to linear time.
• It employs optimized linear algebra techniques to overcome pathological geometries typical in traditional HMC implementations.
• The method enhances effective sample size and integrates seamlessly with probabilistic programming for robust Bayesian inference.

An Analysis of Hamiltonian Monte Carlo for Marginalized Linear Mixed-Effects Models

The paper "Hamiltonian Monte Carlo Inference of Marginalized Linear Mixed-Effects Models" by Lai, Sheldon, and Domke presents an innovative approach to efficiently perform Bayesian inference in Linear Mixed-Effects Models (LMMs) using Hamiltonian Monte Carlo (HMC). LMMs are hierarchical models widely deployed across disciplines such as ecology, medicine, psychology, and neuroscience to account for complex relationships in data by incorporating both fixed and random effects. Traditional HMC implementations often face efficiency challenges due to pathological geometries, such as the funnel shape created by correlation between variance parameters and fixed/random effect parameters. The key contribution of this research is an algorithm that utilizes fast linear algebra techniques to marginalize random effects analytically, reducing computational complexity from cubic to linear time—a significant advancement for practical inference in LMMs.

The authors demonstrate that marginalization mitigates pathologies hindering efficient sampling, substantially improves the efficiency of HMC, and consistently enhances effective sample size (ESS) relative to computation time. These benefits hold particularly for models rooted in cognitive sciences where complex cross-effects models may exist. For instance, models where each observation belongs to multiple random effects groups (e.g., subject and item in lesson evaluations) can significantly benefit from the proposed approach as marginalizing certain groups leads to improved efficiency without degrading sampling quality.

Moreover, the paper systematically addresses challenges of naive marginalization that typically lead to prohibitively expensive cubic time evaluations during HMC iterations, particularly arising from dense covariance matrices. By employing matrix inversion and determinant lemmas optimized for block-diagonal structures, the authors provide a nuanced solution that retains linear time complexity even for relatively high dimensions of random effects in LMMs while ensuring that performance on models with scaled identity covariance matrices—often a practical assumption—is substantially enhanced.

In the Bayesian framework, precise posterior inference is crucial, and the authors’ method interplays directly with commonplace probabilistic programming languages, enhancing user accessibility while automating marginalization steps that are non-trivial for typical users to perform. The integration within platforms such as NumPyro, which is leveraged in their implementation, represents a notable improvement, allowing practitioners to express complex hierarchical models succinctly and rely on backend optimizations for inference efficiency.

The application of the technique is not limited to specific forms of likelihoods but extends to normal and log-normal settings, and potentially other continuous extensions, underscoring its adaptability in broader analytical contexts. As future work, the exploration of broader types of distributions, such as those commonly seen in probit regression or classification tasks, without inducing undesirable computational burdens, can accentuate the versatility and applicability of their methods across diverse inferential scenarios.

This research opens pathways towards integrating marginalization automatically into workflow in probabilistic programming environments, offering potential automation of transformation processes for users with specified LMMs. Future development might aim to routinely apply transformations autonomously, blending user-specified high-level model descriptions with targeted vectorization to further harness the advancements demonstrated.

Overall, this paper presents a significant methodological enhancement in computational Bayesian inference, delivering both theoretical elegance and practical utility, promising enriched capabilities for researchers dealing with the complexities of LMMs and extending Bayesian hierarchical modeling frontiers.