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Model uncertainty and variable selection in Bayesian lasso regression

机译：贝叶斯套索回归中的模型不确定性和变量选择

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摘要

While Bayesian analogues of lasso regression have become popular, comparatively little has been said about formal treatments of model uncertainty in such settings. This paper describes methods that can be used to evaluate the posterior distribution over the space of all possible regression models for Bayesian lasso regression. Access to the model space posterior distribution is necessary if model-averaged inference-e.g., model-averaged prediction and calculation of posterior variable inclusion probabilities-is desired. The key element of all such inference is the ability to evaluate the marginal likelihood of the data under a given regression model, which has so far proved difficult for the Bayesian lasso. This paper describes how the marginal likelihood can be accurately computed when the number of predictors in the model is not too large, allowing for model space enumeration when the total number of possible predictors is modest. In cases where the total number of possible predictors is large, a simple Markov chain Monte Carlo approach for sampling the model space posterior is provided. This Gibbs sampling approach is similar in spirit to the stochastic search variable selection methods that have become one of the main tools for addressing Bayesian regression model uncertainty, and the adaption of these methods to the Bayesian lasso is shown to be straightforward.
机译：尽管套索回归的贝叶斯类似物已经流行，但在这种情况下对模型不确定性的正式处理却鲜有人说。本文介绍了可用于评估贝叶斯套索回归的所有可能回归模型在空间上的后验分布的方法。如果需要模型平均推断（例如，模型平均预测和后变量包含概率的计算），则需要访问模型空间的后验分布。所有此类推论的关键要素是在给定回归模型下评估数据的边际可能性的能力，迄今为止，这已证明对贝叶斯套索来说是困难的。本文介绍了当模型中的预测变量数量不太大时如何精确计算边际可能性，并在可能的预测变量总数适中时允许模型空间枚举。在可能的预测变量总数很大的情况下，提供了一种简单的马尔可夫链蒙特卡洛方法来对模型空间进行后验。这种Gibbs抽样方法在本质上与随机搜索变量选择方法相似，后者已成为解决贝叶斯回归模型不确定性的主要工具之一，而且这些方法对贝叶斯套索的适应性很简单。

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