Developing New Algebraic Geometric Information Criteria for Monte Carlo Inference and Model Selection in Latent Variable and Missing Data Problems

统计模型的选择是一个极具挑战性的问题。描述标准模型选择问题的一种方法是，人们希望选择一种方法，既能很好地拟合观测数据，又能使模型的复杂性最小化（以避免观测数据的过拟合，并失去模型对其他情况的泛化）。实践中使用了许多标准的模型选择准则，如赤池信息准则（AIC）、贝叶斯信息准则（BIC）或最小描述长度准则（MDL）以及偏差信息准则（DIC）。然而，在许多情况下，由于统计模型和/或数据的复杂性，用于选择模型的标准是无效的，甚至可能无法计算（例如，在存在缺失数据或测量误差的情况下）。最近提出的一种机器学习方法使用代数几何中的奇异性解决方法来解决复杂统计模型中的这些问题。拟议的研究计划有五个目标。每个目标都试图将这些新标准与现代模型选择问题的替代现有解决方案联系起来。在每种情况下，我们相信所提出的研究将允许代数几何方法改进现有的解决方案，并极大地影响统计学家拟合和解释现有模型的方式。我们期望所提议的项目将产生新的计算统计方法，新的常用模型的信息标准，以及对缺失数据的复杂模型的统计行为的新见解。

Statistical model selection is an extremely challenging problem. One way to characterize the standard model selection problem is that one wants to choose a method that fits the observed data well while minimizing the complexity of the model (in order to avoid overfitting the observed data and losing generalization of the model to other situations). Many standard model selection criteria are used in practice, such as Akaike's Information Criterion (AIC), the Bayesian Information Criterion (BIC) or Minimum Description Length (MDL), and the Deviance Information Criterion (DIC). However, in many contexts, the criteria used to choose models are not valid or may not be even be calculable because of the complexity of the statistical model and/or data (e.g. in the presence of missing data or measurement error). A recently proposed approach in machine learning uses the resolution of singularities method from algebraic geometry to address these issues in complex statistical models. The proposed research program has five objectives. Each objective attempts to relate these new criteria to alternative, existing solutions to modern model selection problems. In each situation, we believe that the proposed research will allow for algebraic geometric methods to improve upon the existing solutions and dramatically impact the way that statisticians fit and interpret existing models. We expect that the proposed project will generate new computational statistical methods, new information criteria for commonly used models, and new insight into the statistical behavior of complex models for missing data.