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

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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.