Bayesian Clustering with Outliers and Missing Values

The Bayesian Robust Mixture Model (BRMM) is a fully probabilistic model for grouping realvalued data into a finite number of clusters. The model is robust in the sense that it tolerates outliers in the data and handles missing values, both within the Bayesian inference framework. Foreword The purpose of this report is to provide a detailed, step-by-step derivation of the variational update equations for the Bayesian Robust Mixture Model (BRMM). Essential background material on probability distributions an variational approximations is provided in section 1. 1 Background 1.1 Probability distributions This sub-section summarizes definitions and basic properties of the probability distributions we will need for the model. Please note that most of these distributions are parameterized differently than in the literature. My departure from conventional parameterizations will allow for a more intuitive interpretation of the model’s hyper-parameters. For convenience, I have summarized them in table 1. The categorical family A categorical variable is a random variable that takes values in a finite set. Let π = (π1, . . . , πm) be a vector on the standard m− 1 simplex, ∆m−1 = { x = (x1, . . . , xm) ∈ R : 0 ≤ xi ≤ 1 ∀ i, m ∑