Jiaqi Gu and Guosheng Yin’s contribution to the Discussion of ‘Martingale Posterior Distributions’ by Fong, Holmes and Walker
贾琦顾与郭胜尹对《鞅后验分布》的讨论AI 解读 ↓
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贾琦顾与郭胜尹对《鞅后验分布》的讨论
- 研究问题
- 这篇讨论关注鞅后验框架的预测建模与计算特点，并提出如何开展变量选择及联系非参数经验贝叶斯的问题。
- 方法与思路
- 讨论者概述主论文的直接预测建模、数据驱动 copula 参数选择及预测重抽样，再提出错误发现率控制和初始化方向。
- 材料陈述的贡献
- 讨论者强调主论文框架及可并行采样的潜力，并提出变量选择、数据辅助初始化与经验贝叶斯联系；这些是评论与研究建议。
- 阅读关注点
- 应区分讨论者评价与主论文结果，回查预测重抽样、独立性及效率主张的条件，并核对建议方向是否已有实现或理论。
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原文摘要
We congratulate the authors for their thought-provoking paper, especially in the three aspects as follows. Instead of imposing a prior π(θ) on the parameter θ of a likelihood function fθ(y), where both π and fθ are elicited subjectively, the authors proposed the framework of martingale posterior to directly model the predictive p(yn+1:∞∣y1:n) for which only the prior predictive p(y) needs to be specified. As a result, Bayesian inference can be conducted on any parameter (or statistical functional) of the true sampling distribution. As the hyperparameter ρ of the copula is chosen in such a data-driven way that p(y1:n) fits the observed data y1:n well, the impact of the initial guess for the prior predictive, p0(y), on p(yn+1:∞∣y1:n) can be adjusted properly. Unlike existing MCMC methods which generate autocorrelated posterior samples, the proposed predictive resampling algorithm is GPU-friendly and parallelisable. Thus, independent posterior samples can be obtained for inference, leading to improvement of both computational and statistical efficiency. The proposed martingale posterior framework, we believe, has the further potentials in several directions. Because variable selection can generally improve statistical efficiency of regression analysis, it is of interest to investigate how Bayesian variable selection can be conducted with control of the Bayesian false discovery rate under the framework of martingale posterior. Although a default choice is suggested for the initial guess p0(y), it is possible to leverage some information from the observed data y1:n to initialise p0(y), similar to empirical Bayes methods. This raises a question on how the martingale posterior connects to nonparametric empirical Bayes, which also estimates the prior from the data.