Evaluating Sensitivity to the Stick-Breaking Prior in Bayesian Nonparametrics
Evaluating Sensitivity to the Stick-Breaking Prior in Bayesian Nonparametrics
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DOI：
10.1214/22-ba1309
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发表时间：
2018-10
影响因子：
4.4
通讯作者：
Runjing Liu;Ryan Giordano;Michael I. Jordan;Tamara Broderick
中科院分区：
文献类型：
--
作者：
Runjing Liu;Ryan Giordano;Michael I. Jordan;Tamara Broderick
许多概率聚类问题的核心问题是特定数据集中存在多少个不同的聚类。贝叶斯非参数 (BNP) 模型通过在聚类分配上放置生成过程来解决这个问题。然而，与所有贝叶斯方法一样，BNP 需要先验的规范。在实践中，重要的是要定量地确定先验信息不太丰富，特别是当选择先验的特定形式是为了数学方便而不是因为经过考虑的主观信念时。我们推导出截断变分贝叶斯 (VB) 近似的局部灵敏度测量，并使用局部泰勒级数近似来近似 VB 最优值对先验参数的非线性依赖性。使用狄利克雷过程的破棒表示，我们考虑对标量浓度参数和破棒分布的函数形式的扰动。与之前关于 BNP 局部贝叶斯敏感性的工作不同，我们特别关注我们的敏感性测量推断不同先验的能力，而不是将敏感性视为鲁棒性本身的度量。外推法促使对 VB 的先验函数形式使用乘法扰动。此外，我们仅对推理的计算密集部分（全局参数的优化）进行线性近似，并将容易计算的量的非线性保留为全局参数的函数。我们应用我们的方法来估计 Iris 数据集中存在的不同簇的预期数量对 BNP 先验规范的敏感性。我们通过与昂贵得多的重新拟合模型的过程进行比较来评估近似值的准确性。
A central question in many probabilistic clustering problems is how many distinct clusters are present in a particular dataset. A Bayesian nonparametric (BNP) model addresses this question by placing a generative process on cluster assignment. However, like all Bayesian approaches, BNP requires the specification of a prior. In practice, it is important to quantitatively establish that the prior is not too informative, particularly when the particular form of the prior is chosen for mathematical convenience rather than because of a considered subjective belief. We derive local sensitivity measures for a truncated variational Bayes (VB) approximation and approximate nonlinear dependence of a VB optimum on prior parameters using a local Taylor series approximation. Using a stick-breaking representation of a Dirichlet process, we consider perturbations both to the scalar concentration parameter and to the functional form of the stick- breaking distribution. Unlike previous work on local Bayesian sensitivity for BNP, we pay special attention to the ability of our sensitivity measures to extrapolate to different priors, rather than treating the sensitivity as a measure of robustness per se. Extrapolation motivates the use of multiplicative perturbations to the functional form of the prior for VB. Additionally, we linearly approximate only the computationally intensive part of inference -- the optimization of the global parameters -- and retain the nonlinearity of easily computed quantities as functions of the global parameters. We apply our methods to estimate sensitivity of the expected number of distinct clusters present in the Iris dataset to the BNP prior specification. We evaluate the accuracy of our approximations by comparing to the much more expensive process of re-fitting the model.