Being an informed Bayesian: Assessing prior informativeness and prior likelihood conflict
Being an informed Bayesian: Assessing prior informativeness and prior likelihood conflict
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成为知情的贝叶斯主义者：评估先验信息性和先验可能性冲突
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
D. Nicolae
中科院分区：
文献类型：
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作者：
M. Reimherr;X. Meng;D. Nicolae
贝叶斯方法的常规应用大大扩展，大大增加了评估先验分布中关于我们的似然函数提供的信息的确认性和矛盾性信息的需要。我们提出了一种从熟悉的后验匹配法开始的诊断方法。对于给定的似然模型，我们确定形成两个似然函数所需的信息差异，当这两个似然函数分别与给定的先验和基线先验相结合时，将导致相同的后验不确定性。在具有独立的、相同分布的样本的情况下，样本大小是信息的自然度量，该差异可以被视为关于基于$k$观测的似然函数的先验数据大小$M(K)$。当相对于基线没有可检测到的先验可能性冲突时，$M(K)$在$k$上大致不变，这是一个捕捉确凿信息的常量。否则，$M(K)$倾向于随着$k$而减少，因为矛盾的先验从似然函数中减去了信息。在极端矛盾的情况下，$M(K)/k$将接近其下界$-1$，表示由于冲突而完全取消先验信息和可能性信息。我们还报告了一个有趣的超级信息量现象，当先验平均值与事实相符时，先验数据大小相对于其名义大小有效地获得了额外的$(1+r)^{-1}$%，其中$r$是名义先前数据大小相对于后验数据大小的百分比。我们通过几个例子来演示我们的方法，包括探索免疫球蛋白水平对狼疮性肾炎的影响的应用程序。我们还为我们的方法提供了一个理论基础，几乎所有的似然先验对都具有渐近共轭。
Dramatically expanded routine adoption of the Bayesian approach has substantially increased the need to assess both the confirmatory and contradictory information in our prior distribution with regard to the information provided by our likelihood function. We propose a diagnostic approach that starts with the familiar posterior matching method. For a given likelihood model, we identify the difference in information needed to form two likelihood functions that, when combined respectively with a given prior and a baseline prior, will lead to the same posterior uncertainty. In cases with independent, identically distributed samples, sample size is the natural measure of information, and this difference can be viewed as the prior data size $M(k)$, with regard to a likelihood function based on $k$ observations. When there is no detectable prior-likelihood conflict relative to the baseline, $M(k)$ is roughly constant over $k$, a constant that captures the confirmatory information. Otherwise $M(k)$ tends to decrease with $k$ because the contradictory prior detracts information from the likelihood function. In the case of extreme contradiction, $M(k)/k$ will approach its lower bound $-1$, representing a complete cancelation of prior and likelihood information due to conflict. We also report an intriguing super-informative phenomenon where the prior effectively gains an extra $(1+r)^{-1}$ percent of prior data size relative to its nominal size when the prior mean coincides with the truth, where $r$ is the percentage of the nominal prior data size relative to the total data size underlying the posterior. We demonstrate our method via several examples, including an application exploring the effect of immunoglobulin levels on lupus nephritis. We also provide a theoretical foundation of our method for virtually all likelihood-prior pairs that possess asymptotic conjugacy.