A Bayesian Filtering Algorithm for Gaussian Mixture Models

A Bayesian filtering algorithm is developed for a class of state-space systems that can be modelled via Gaussian mixtures. In general, the exact solution to this filtering problem involves an exponential growth in the number of mixture terms and this is handled here by utilising a Gaussian mixture reduction step after both the time and measurement updates. In addition, a square-root implementation of the unified algorithm is presented and this algorithm is profiled on several simulated systems. This includes the state estimation for two non-linear systems that are strictly outside the class considered in this paper

Incremental Learning of Nonparametric Bayesian Mixture Models

Clustering is a fundamental task in many vision applications. To date, most clustering algorithms work in a batch setting and training examples must be gathered in a large group before learning can begin. Here we explore incremental clustering, in which data can arrive continuously. We present a novel incremental model-based clustering algorithm based on nonparametric Bayesian methods, which we call Memory Bounded Variational Dirichlet Process (MB-VDP). The number of clusters are determined flexibly by the data and the approach can be used to automatically discover object categories. The computational requirements required to produce model updates are bounded and do not grow with the amount of data processed. The technique is well suited to very large datasets, and we show that our approach outperforms existing online alternatives for learning nonparametric Bayesian mixture models

A Tutorial on Bayesian Nonparametric Models

A key problem in statistical modeling is model selection, how to choose a model at an appropriate level of complexity. This problem appears in many settings, most prominently in choosing the number ofclusters in mixture models or the number of factors in factor analysis. In this tutorial we describe Bayesian nonparametric methods, a class of methods that side-steps this issue by allowing the data to determine the complexity of the model. This tutorial is a high-level introduction to Bayesian nonparametric methods and contains several examples of their application.Comment: 28 pages, 8 figure

How Many Communities Are There?

Stochastic blockmodels and variants thereof are among the most widely used approaches to community detection for social networks and relational data. A stochastic blockmodel partitions the nodes of a network into disjoint sets, called communities. The approach is inherently related to clustering with mixture models; and raises a similar model selection problem for the number of communities. The Bayesian information criterion (BIC) is a popular solution, however, for stochastic blockmodels, the conditional independence assumption given the communities of the endpoints among different edges is usually violated in practice. In this regard, we propose composite likelihood BIC (CL-BIC) to select the number of communities, and we show it is robust against possible misspecifications in the underlying stochastic blockmodel assumptions. We derive the requisite methodology and illustrate the approach using both simulated and real data. Supplementary materials containing the relevant computer code are available online.Comment: 26 pages, 3 figure