Sampling from Density power divergence-based Generalized posterior distribution via Stochastic optimization (2501.07790v1)

Abstract: Robust Bayesian inference using density power divergence (DPD) has emerged as a promising approach for handling outliers in statistical estimation. While the DPD-based posterior offers theoretical guarantees for robustness, its practical implementation faces significant computational challenges, particularly for general parametric models with intractable integral terms. These challenges become especially pronounced in high-dimensional settings where traditional numerical integration methods prove inadequate and computationally expensive. We propose a novel sampling methodology that addresses these limitations by integrating the loss-likelihood bootstrap with a stochastic gradient descent algorithm specifically designed for DPD-based estimation. Our approach enables efficient and scalable sampling from DPD-based posteriors for a broad class of parametric models, including those with intractable integrals, and we further extend it to accommodate generalized linear models. Through comprehensive simulation studies, we demonstrate that our method efficiently samples from DPD-based posteriors, offering superior computational scalability compared to conventional methods, particularly in high-dimensional settings. The results also highlight its ability to handle complex parametric models with intractable integral terms.

• The paper introduces a novel SGD-based method that bypasses intractable integrals in density power divergence formulations.
• It demonstrates superior computational efficiency and robustness to outliers compared to traditional MCMC techniques.
• Empirical validation shows its scalable performance in high-dimensional generalized linear models.

Sampling from Density Power Divergence-Based Generalized Posterior Distribution via Stochastic Optimization

The paper introduces a novel methodology for sampling from density power divergence (DPD)-based generalized posterior distributions, addressing significant computational challenges typically associated with robust Bayesian inference. The research prioritizes two key obstacles: the intractability of integral terms in DPD formulations for general parametric models and the excessive computational burden in high-dimensional settings. Leveraging a stochastic gradient-based optimization framework, the authors propose a method that adeptly balances robustness against outliers while ensuring computational efficiency, a critical requirement in contemporary high-dimensional data analysis.

Methodological Innovations

This paper's core contribution is the development of a stochastic gradient descent (SGD)-based sampling approach, which efficiently handles the complexities inherent in DPD-based posterior distributions. Traditional methods, relying on MCMC techniques such as Metropolis-Hastings (MH) or even Hamiltonian Monte Carlo (HMC), often encounter scalability issues when dealing with models involving intractable integrals or high-dimensional data spaces. By contrast, the proposed SGD approach integrates the loss-likelihood bootstrap into its framework, resulting in a method capable of generating posterior samples without necessitating comprehensive numerical integration of these intractable terms.

The approach is not only mathematically innovative but also practical. It circumvents the need for explicit computation of the DPD integral term via stochastic approximation, thereby substantially reducing the computational load. This makes the method particularly appealing for generalized linear models (GLMs), a category where conventional numerical integration strategies become untenable due to the dimensionality of the parameter space.

Empirical Validation

The authors substantiate their methodology through extensive simulation studies that compare their approach with standard sampling algorithms, including MH and traditional LLB with numerical integration. Two primary assessment metrics stand out: computational efficiency and robustness to outliers. Specifically, the proposed method demonstrates superior scalability, maintaining stable computational times even as model dimensions increase—a stark contrast to the exponential growth seen with traditional methods. Furthermore, the empirical results underscore the method's robustness, as it maintains high accuracy in parameter estimation despite the presence of data contamination.

Practical and Theoretical Implications

Practically, this research offers a scalable solution for practitioners involved in high-dimensional robust Bayesian inference. The method's ability to sample efficiently from complex posterior distributions means that it can handle real-world applications requiring robust estimation against outliers and noise, without succumbing to the computational drawbacks endemic to conventional techniques. Theoretically, the paper advances the understanding of stochastic gradient methods' utility in Bayesian inference, expanding the horizon of model types and dimensionality that robust methods can feasibly address.

Future Prospects

Looking forward, there are several avenues for enriching this research. Extending the framework to integrate informative prior distributions could enhance the model's applicability in settings where prior information can significantly aid estimation accuracy. Furthermore, expanding the methodology to accommodate hierarchical structures inherent in generalized linear mixed models remains an open and promising direction. The potential to explore the effect of posterior predictive stability, particularly in the context of DPD-induced corrections to model misspecification, presents another notable opportunity for expansion.

In summation, this work marks a significant stride in addressing the computational barriers of high-dimensional and intractable models within robust Bayesian frameworks. Through meticulous methodological development and comprehensive validation, the authors present a tool both versatile and efficient, equipped to meet the growing demands of modern statistical analysis.