Detecting Latent Structure Uncertainty with Structural Entropy

This paper proposes a new method for detecting the uncertainty of a latent structure. We consider the case where the latent structure of dataset changes gradually over time, with the goal of selecting the optimal model at any given time. In selecting the optimal model, we use the minimum description length (MDL) principle, specifically the normalized maximum likelihood (NML), which is the optimal code length in the sense of Shtarkov's minimax regret. To detect the uncertainty of a latent structure, the main idea proposed here is that the uncertainty of the model selection will increase at the initial stage when the model change occurs. Here, we propose a new indicator called "structural entropy (SE)", which defines model selection uncertainty based on the MDL principle. We use several models for model selection, including the clustering structures of the Gaussian mixture model and Poisson mixture model, and a time-dependent structure model such as the autoregression model. We show the behavior of the proposed indicator (SE) using an artificial dataset and a real marketing dataset.