Conjugate Bayes for probit regression via unified skew-normal distributions
通过统一偏斜正态分布实现 probit 回归的共轭贝叶斯推断AI 解读 ↓
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通过统一偏斜正态分布实现 probit 回归的共轭贝叶斯推断
- 研究问题
- 如何刻画高斯先验下 probit 系数的后验分布，并在高维小样本情形改善计算，同时保留后验分布的偏斜特征？
- 方法与思路
- 作者证明高斯先验下 probit 系数后验具有统一偏斜正态核，据此开展贝叶斯推断，并发展共轭先验及独立同分布抽样方法。
- 材料陈述的贡献
- 摘要称，该后验表示支持高维、较小样本下的高效推断，避免忽略偏斜特征，并通过遗传研究展示相应方法。
- 阅读关注点
- 应核对后验分布的参数映射、共轭先验类别、独立抽样的具体算法，以及计算优势在不同维度和样本量下的证据。
- 阅读边界
- 仅据 Crossref 摘要；具体计算代价与比较数值未说明，所述结果针对 probit，不能自动推广到 logit 或所有二元模型。
原文摘要
Summary Regression models for dichotomous data are ubiquitous in statistics. Besides being useful for inference on binary responses, these methods serve as building blocks in more complex formulations, such as density regression, nonparametric classification and graphical models. Within the Bayesian framework, inference proceeds by updating the priors for the coefficients, typically taken to be Gaussians, with the likelihood induced by probit or logit regressions for the responses. In this updating, the apparent absence of a tractable posterior has motivated a variety of computational methods, including Markov chain Monte Carlo routines and algorithms that approximate the posterior. Despite being implemented routinely, Markov chain Monte Carlo strategies have mixing or time-inefficiency issues in large-$p$ and small-$n$ studies, whereas approximate routines fail to capture the skewness typically observed in the posterior. In this article it is proved that the posterior distribution for the probit coefficients has a unified skew-normal kernel under Gaussian priors. This result allows efficient Bayesian inference for a wide class of applications, especially in large-$p$ and small-to-moderate-$n$ settings where state-of-the-art computational methods face notable challenges. These advances are illustrated in a genetic study, and further motivate the development of a wider class of conjugate priors for probit models, along with methods for obtaining independent and identically distributed samples from the unified skew-normal posterior.