Abstract

We consider the Jeffreys-Lindley paradox from an objective Bayesian perspective by attempting to find priors representing complete indifference to sample size in the problem. This means that we ensure that the prior for the unknown mean and the prior predictive for the t-statistic are independent of the sample size. If successful, this would lead to Bayesian model comparison that was independent of sample size and ameliorate the paradox. Unfortunately, it leads to an improper scale-invariant prior for the unknown mean. We show, however, that a truncated scale-invariant prior delays the dependence on sample size, which could be practically significant. Lastly, we shed light on the paradox by relating it to the fact that the scale-invariant prior is improper.

摘要

我们从客观贝叶斯的角度考虑 Jeffreys-Lindley 悖论，试图找到代表问题中样本大小完全无差异的先验。这意味着我们确保未知均值的先验和 t 统计量的先验预测与样本量无关。如果成功，这将导致与样本量无关的贝叶斯模型比较并改善悖论。不幸的是，它导致未知均值的先验比例不变量不正确。然而，我们表明，截断的尺度不变先验延迟了对样本大小的依赖，这实际上可能很重要。最后，我们通过将其与尺度不变先验不正确的事实联系起来来阐明这个悖论。