1. The Bayesian information criterion (BIC) has been proposed as a way to carry out Bayesian hypothesis testing when there are no clear expectations. However, the BIC rests on a particular prior distribution, for which there is rarely any justification. See Raftery (1995) on the case for the BIC and Weakliem (1999) for a critique. 2. The assumption that the sample is of the same size is important. To obtain the expected prediction error in a sample of arbitrary size, it is necessary to know the true model. Consequently, there is no method of model selection that uniformly leads to better out-of-sample predictions. 3. Schultz proposes that the value should be exp(AIC2 – AIC1), or about .0025 in this example. I think this is mistaken, and it should be exp{(AIC2 – AIC1)/2}. The general point about considering the theoretical probability of a nonzero value applies regardless of which formula is correct.

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评论：贝叶斯视角下的推理信息准则

1. 贝叶斯信息准则 (BIC) 已被提出作为一种在没有明确期望时进行贝叶斯假设检验的方法。然而，BIC 依赖于特定的先验分布，对此很少有任何理由。参见 Raftery (1995) 关于 BIC 的案例和 Weakliem (1999) 的评论。2. 样本大小相同的假设很重要。要在任意大小的样本中获得预期的预测误差，必须知道真实模型。因此，没有一种模型选择方法可以统一导致更好的样本外预测。3. Schultz 建议该值应为 exp(AIC2 – AIC1)，在本例中约为 0.0025。我认为这是错误的，应该是 exp{(AIC2 – AIC1)/2}。