量化观察到的既往影响
Quantifying Observed Prior Impact
摘要：
我们区分两个问题：（i）先验包含多少信息？及（ii）该优先权的效力为何？已经提出了几种措施来量化有效的先验样本量，例如Clarke [1996]和Morita等人[2008]。然而，这些措施通常忽略了可能性的推断目前手头上，因此解决（i）而不是（ii）。由于在实践中（ii）非常值得关注，Reimherr等人[2014]基于先验似然不一致性引入了一类新的有效先验样本量测量。我们把这个想法进一步走向其自然贝叶斯的结论，提出措施的有效先验样本量，不仅纳入了一般的数学形式的可能性，但也手头的具体数据。因此，我们的测量并不是在工作模型的数据集上取平均值，而是以当前观察到的数据为条件。因此，我们的措施可以是高度可变的，但我们证明，这是因为先验的影响可以是高度可变的。我们的措施是贝叶斯估计的有意义的数量和沟通的程度，推断是由先验，或不同的框架，节省了大量的努力，由于有先验信息。我们通过一些例子来说明我们的想法，包括高斯共轭模型（连续观测），贝塔二项式模型（离散观测），和线性回归模型（两个未知参数）。最后讨论了该方法的进一步发展和在天文学中的应用的未来工作。
英文摘要：
We distinguish two questions (i) how much information does the prior contain? and (ii) what is the effect of the prior? Several measures have been proposed for quantifying effective prior sample size, for example Clarke [1996] and Morita et al. [2008]. However, these measures typically ignore the likelihood for the inference currently at hand, and therefore address (i) rather than (ii). Since in practice (ii) is of great concern, Reimherr et al. [2014] introduced a new class of effective prior sample size measures based on prior-likelihood discordance. We take this idea further towards its natural Bayesian conclusion by proposing measures of effective prior sample size that not only incorporate the general mathematical form of the likelihood but also the specific data at hand. Thus, our measures do not average across datasets from the working model, but condition on the current observed data. Consequently, our measures can be highly variable, but we demonstrate that this is because the impact of a prior can be highly variable. Our measures are Bayes estimates of meaningful quantities and well communicate the extent to which inference is determined by the prior, or framed differently, the amount of effort saved due to having prior information. We illustrate our ideas through a number of examples including a Gaussian conjugate model (continuous observations), a Beta-Binomial model (discrete observations), and a linear regression model (two unknown parameters). Future work on further developments of the methodology and an application to astronomy are discussed at the end.
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