Theory of Dependent Hierarchical Normalized Random Measures

TL;DR

This paper develops a comprehensive theoretical framework for dependent hierarchical normalized random measures, including NGGs, with applications in time-dependent topic modeling and posterior inference methods.

Contribution

It introduces mathematical foundations, dependency operators, and inference techniques for hierarchical NRMs and NGGs, advancing their analysis and application in dependent network models.

Findings

Developed dependency and composition results for hierarchical NRMs.

Introduced slice sampling for posterior inference of NGGs.

Provided theoretical tools for analyzing networks of dependent NRMs.

Abstract

This paper presents theory for Normalized Random Measures (NRMs), Normalized Generalized Gammas (NGGs), a particular kind of NRM, and Dependent Hierarchical NRMs which allow networks of dependent NRMs to be analysed. These have been used, for instance, for time-dependent topic modelling. In this paper, we first introduce some mathematical background of completely random measures (CRMs) and their construction from Poisson processes, and then introduce NRMs and NGGs. Slice sampling is also introduced for posterior inference. The dependency operators in Poisson processes and for the corresponding CRMs and NRMs is then introduced and Posterior inference for the NGG presented. Finally, we give dependency and composition results when applying these operators to NRMs so they can be used in a network with hierarchical and dependent relations.