Leach Justin M, Yi Nengjun, Aban Inmaculada
Department of Biostatistics, University of Alabama at Birmingham, Birmingham, AL, USA.
J Appl Stat. 2023 Sep 14;51(11):2039-2061. doi: 10.1080/02664763.2023.2258301. eCollection 2024.
Spike-and-slab prior distributions are used to impose variable selection in Bayesian regression-style problems with many possible predictors. These priors are a mixture of two zero-centered distributions with differing variances, resulting in different shrinkage levels on parameter estimates based on whether they are relevant to the outcome. The spike-and-slab lasso assigns mixtures of double exponential distributions as priors for the parameters. This framework was initially developed for linear models, later developed for generalized linear models, and shown to perform well in scenarios requiring sparse solutions. Standard formulations of generalized linear models cannot immediately accommodate categorical outcomes with > 2 categories, i.e. multinomial outcomes, and require modifications to model specification and parameter estimation. Such modifications are relatively straightforward in a Classical setting but require additional theoretical and computational considerations in Bayesian settings, which can depend on the choice of prior distributions for the parameters of interest. While previous developments of the spike-and-slab lasso focused on continuous, count, and/or binary outcomes, we generalize the spike-and-slab lasso to accommodate multinomial outcomes, developing both the theoretical basis for the model and an expectation-maximization algorithm to fit the model. To our knowledge, this is the first generalization of the spike-and-slab lasso to allow for multinomial outcomes.
尖峰和平板先验分布用于在具有许多可能预测变量的贝叶斯回归式问题中进行变量选择。这些先验是两个具有不同方差的零中心分布的混合，根据参数是否与结果相关，对参数估计产生不同程度的收缩。尖峰和平板套索将双指数分布的混合作为参数的先验。该框架最初是为线性模型开发的，后来扩展到广义线性模型，并在需要稀疏解的场景中表现良好。广义线性模型的标准公式不能直接处理类别数大于2的分类结果，即多项结果，需要对模型规范和参数估计进行修改。在经典环境中，这种修改相对简单，但在贝叶斯环境中需要额外的理论和计算考虑，这可能取决于对感兴趣参数的先验分布的选择。虽然之前尖峰和平板套索的发展集中在连续、计数和/或二元结果上，但我们将尖峰和平板套索进行了推广以适应多项结果，开发了模型的理论基础和一种期望最大化算法来拟合模型。据我们所知，这是首次将尖峰和平板套索推广到允许多项结果的情况。
