Bayesian Estimation of Multidimensional Latent Variables and Its Asymptotic Accuracy
This work addresses a key issue in unsupervised learning for researchers and practitioners, but it is incremental as it builds on prior algebraic geometry-based methods.
The paper tackles the problem of analyzing the accuracy of latent variable estimation in hierarchical learning models, extending a previous method to handle latent variables with redundant dimensions and deriving asymptotic error functions.
Hierarchical learning models, such as mixture models and Bayesian networks, are widely employed for unsupervised learning tasks, such as clustering analysis. They consist of observable and hidden variables, which represent the given data and their hidden generation process, respectively. It has been pointed out that conventional statistical analysis is not applicable to these models, because redundancy of the latent variable produces singularities in the parameter space. In recent years, a method based on algebraic geometry has allowed us to analyze the accuracy of predicting observable variables when using Bayesian estimation. However, how to analyze latent variables has not been sufficiently studied, even though one of the main issues in unsupervised learning is to determine how accurately the latent variable is estimated. A previous study proposed a method that can be used when the range of the latent variable is redundant compared with the model generating data. The present paper extends that method to the situation in which the latent variables have redundant dimensions. We formulate new error functions and derive their asymptotic forms. Calculation of the error functions is demonstrated in two-layered Bayesian networks.