Bayesian Methods for Data-Dependent Priors
Bayesian Methods for Data-Dependent Priors
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发表时间：
2011
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通讯作者：
William Francis Darnieder
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作者：
William Francis Darnieder
在贝叶斯统计建模中，数据依赖先验的使用奇怪地无处不在，因为它在一些干部中引起了警觉。事实上，采用数据依赖先验的研究者面临着一个不可避免的事实，即他或她已经使用了两次数据：第一次是建立先验信念，第二次是用可能性更新先验以获得后验分布。在这篇论文中，我们介绍了调整后的数据依赖贝叶斯范式作为一个原则性的方法，使用数据依赖先验弱符合贝叶斯定理。假设研究人员在应用贝叶斯更新之前通过一些汇总统计量来查看数据。这种新颖的方法系统地选择后验分布，该后验分布捕获关于包含在数据中的模型参数的信息，该信息超出了包含在观察到的统计量中的信息。在特殊情况下，调整后的方法在形式上等同于其他贝叶斯方法，但这些情况必然很少。调整后的过程施加一个空更新时，观察到的统计是足够的，选择后验分布，等于前。相反，观察到不充分的统计数据将导致不平凡的更新。特别令人感兴趣的是如何在调整和天真（未调整）程序下进行分析比较。实施策略进行了描述，在低和高维设置的调整。后验模拟强调ii使用马尔可夫链蒙特卡罗（MCMC）技术修改，以适应调整后的范例。黑箱策略（如预处理数据）用于适应狄利克雷过程（DP）混合模型被铸造为数据依赖，我们演示了如何在这些设置中应用调整。此外，我们比较了在分析Roeder（1990）推广的经典星系数据集时，朴素和调整技术的预测能力。
The use of data-dependent priors is strangely ubiquitous in Bayesian statistical modeling given the alarm it raises within some cadres. Indeed, the researcher who employs a data-dependent prior faces the inescapable truth that he or she has used the data twice: the first time in establishing prior belief, and the second in updating the prior with the likelihood to obtain the posterior distribution. In this dissertation, we introduce the Adjusted Data-Dependent Bayesian Paradigm as a principled approach to using data-dependent priors in weak accordance with Bayes’ theorem. Suppose that a researcher peeks at the data through some summary statistic prior to applying the Bayesian update. This novel method systematically chooses the posterior distribution that captures the information regarding model parameters that is contained in the data above and beyond the information contained in the observed statistic. In special situations the adjusted approach is formally equivalent to other Bayesian methods, but these cases are necessarily rare. The adjusted procedure imposes a null update when the observed statistic is sufficient, choosing the posterior distribution that is equal to the prior. Conversely, observing a non-sufficient statistic will invite a non-trivial update. Of particular interest is how analyses under the adjusted and naive (unadjusted) procedures compare. Implementation strategies are described for imposing the adjustment in low and high dimensional settings. Posterior simulation is emphasized ii using Markov Chain Monte Carlo (MCMC) techniques modified to accommodate the adjusted paradigm. Black box strategies (e.g. preprocessing data) used to fit Dirichlet Process (DP) mixture models are cast as data-dependent and we demonstrate how to apply the adjustment in these settings. Additionally, we compare the predictive power of the naive and adjusted techniques in analyzing the classic galaxies data set popularized in Roeder (1990).