Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN 47405-7007, USA.
Br J Math Stat Psychol. 2013 Feb;66(1):45-56. doi: 10.1111/j.2044-8317.2012.02063.x. Epub 2012 Sep 24.
Bayesian inference is conditional on the space of models assumed by the analyst. The posterior distribution indicates only which of the available parameter values are less bad than the others, without indicating whether the best available parameter values really fit the data well. A posterior predictive check is important to assess whether the posterior predictions of the least bad parameters are discrepant from the actual data in systematic ways. Gelman and Shalizi (2012a) assert that the posterior predictive check, whether done qualitatively or quantitatively, is non-Bayesian. I suggest that the qualitative posterior predictive check might be Bayesian, and the quantitative posterior predictive check should be Bayesian. In particular, I show that the 'Bayesian p-value', from which an analyst attempts to reject a model without recourse to an alternative model, is ambiguous and inconclusive. Instead, the posterior predictive check, whether qualitative or quantitative, should be consummated with Bayesian estimation of an expanded model. The conclusion agrees with Gelman and Shalizi regarding the importance of the posterior predictive check for breaking out of an initially assumed space of models. Philosophically, the conclusion allows the liberation to be completely Bayesian instead of relying on a non-Bayesian deus ex machina. Practically, the conclusion cautions against use of the Bayesian p-value in favour of direct model expansion and Bayesian evaluation.
贝叶斯推断取决于分析师所假设的模型空间。后验分布仅表明可用参数值中哪些比其他值更差,而不表明可用的最佳参数值是否真的很好地拟合数据。后验预测检查对于评估后验预测中最差参数是否以系统方式与实际数据存在差异非常重要。Gelman 和 Shalizi(2012a)断言,后验预测检查无论是定性的还是定量的,都不是贝叶斯的。我认为定性后验预测检查可能是贝叶斯的,而定量后验预测检查应该是贝叶斯的。特别是,我表明,分析师试图在不诉诸替代模型的情况下拒绝模型的“贝叶斯 p 值”是模糊和不确定的。相反,后验预测检查,无论是定性的还是定量的,都应该与扩展模型的贝叶斯估计相结合完成。该结论与 Gelman 和 Shalizi 关于后验预测检查对于突破初始假设模型空间的重要性的观点一致。从哲学上讲,该结论允许完全采用贝叶斯方法,而不是依赖于非贝叶斯的解围神。从实践上讲,该结论警告不要使用贝叶斯 p 值,而应直接扩展模型并进行贝叶斯评估。