Bibliographic Information: Dolera, E., Favaro, S., & Giordano, M. (2024). On strong posterior contraction rates for Besov-Laplace priors in the white noise model. arXiv preprint arXiv:2411.06981v1.
Research Objective: This paper investigates the asymptotic performance of smoothness-matching Besov-Laplace priors in the white noise model, aiming to determine if they can achieve minimax optimal posterior contraction rates for spatially inhomogeneous functions and their derivatives.
Methodology: The authors utilize a novel "Wasserstein dynamics" approach to analyze posterior contraction rates. This method leverages the properties of Wasserstein distance, Lipschitz continuity of posterior distributions, Poincaré constant estimates for log-concave measures, and the Laplace method for integral approximation.
Key Findings: The study reveals that smoothness-matching Besov-Laplace priors do indeed attain minimax optimal posterior contraction rates in strong Sobolev metrics for ground truths belonging to Besov spaces. This finding challenges previous research that advocated for undersmoothing and rescaling of Besov priors to achieve optimality. Furthermore, the authors demonstrate that these priors possess a desirable plug-in property for derivative estimation, implying that the push-forward measures under differential operators can optimally recover the derivatives of the unknown function.
Main Conclusions: The research concludes that smoothness-matching Besov-Laplace priors offer a powerful and elegant approach for estimating spatially inhomogeneous functions and their derivatives in the white noise model. The study highlights the effectiveness of the Wasserstein dynamics approach in overcoming the limitations of traditional testing-based methods for analyzing posterior contraction rates.
Significance: This work significantly contributes to the field of Bayesian nonparametric inference by providing theoretical justification for using smoothness-matching Besov-Laplace priors. It offers a more natural and potentially computationally advantageous alternative to the previously employed undersmoothing and rescaling techniques.
Limitations and Future Research: The study primarily focuses on the white noise model. Future research could explore the applicability of these findings to other statistical models and investigate the performance of smoothness-matching Besov-Laplace priors in more complex settings. Additionally, exploring adaptive priors within this framework could further enhance the flexibility and applicability of this approach.
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by Emanuele Dol... at arxiv.org 11-12-2024
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