Abstract

The problem considered is that of making inferences about the value of a parameter vector $\mathit{\theta }$ based on the value of an observable random vector y that is subject to selection of the form $\mathbf{y}\in S$ (for a known subset S). According to conventional wisdom, a Bayesian approach (unlike a frequentist approach) requires no adjustment for selection, which is generally regarded as counterintuitive and even paradoxical. An alternative considered herein consists (when taking a Bayesian approach in the face of selection) of basing the inferences for the value of $\mathit{\theta }$ on the posterior distribution derived from the conditional (on $\mathbf{y}\in S$) joint distribution of y and $\mathit{\theta }$. That leads to an adjustment in the likelihood function that is reinterpretable as an adjustment to the prior distribution and ultimately leads to a different posterior distribution. And it serves to make the inferences specific to settings that are subject to selection of the same kind as the setting that gave rise to the data. Moreover, even in the absence of any real selection, this approach can be used to make the inferences specific to a meaningful subset of y-values.

摘要

考虑的问题是对参数向量的值进行推断$\mathit{\theta }$基于可观察到的随机向量 y 的值，该值受形式选择的影响$\mathbf{是的}\in \mathrm{小号}$（对于已知的子集S）。根据传统观点，贝叶斯方法（与频率论方法不同）不需要对选择进行调整，这通常被认为是违反直觉的，甚至是自相矛盾的。这里考虑的一个替代方案包括（当在面对选择时采用贝叶斯方法时）基于$\mathit{\theta }$关于从条件导出的后验分布（在$\mathbf{是的}\in \mathrm{小号}$) y 的联合分布和$\mathit{\theta }$. 这导致似然函数的调整可以重新解释为对先验分布的调整，并最终导致不同的后验分布。并且它用于对与产生数据的设置相同类型的设置进行特定的推论。此外，即使在没有任何实际选择的情况下，这种方法也可用于对有意义的 y 值子集进行特定的推断。