Introduction

Clustering is a fundamental technique in data analysis, where the objective is to group a set of objects in such a way that objects in the same group, called a cluster, are more similar than those in other groups. Unsupervised clustering with a Dirichlet Process (DP) is a powerful approach to infer the number of clusters in the data instead of predefining them. In this article, we will explore how to extract unsupervised clusters using a Dirichlet Process in PyMC3, a Python library for Bayesian statistical modeling.

Understanding Dirichlet Processes

What is a Dirichlet Process?

The Dirichlet Process is a stochastic process used in Bayesian nonparametric models. It is particularly useful for problems where the number of mixture components (clusters) is unknown and is instead treated as a random variable. The amount of flexibility allowed by DPs makes them a potent tool for clustering problems.

Mathematically, a Dirichlet Process is a distribution over distributions. It is characterized by a base distribution $G_0$ and a concentration parameter $\alpha$. The concentration parameter indicates how strongly the DP tends to concentrate on the base distribution — a high value of $\alpha$ means more clusters, while a low value tends to produce fewer clusters.

Chinese Restaurant Process

The Dirichlet Process can be visualized through the analogy of a Chinese Restaurant Process (CRP). Imagine an infinitely long table in a restaurant. The first customer sits at the first table; subsequent customers either join an existing table with a probability proportional to the number of people already sitting there or start a new table with a probability proportional to $\alpha$.

Implementing Dirichlet Process Clustering in PyMC3

PyMC3 provides the tools needed to model a Dirichlet Process. The following example demonstrates how to implement unsupervised clustering using a DP in PyMC3.

• We define a PyMC3 model which includes a DP. The hyperparameter $\alpha$ is modeled with a Gamma distribution, which provides the base concentration for the model.
• A stick-breaking process is employed. Here, each component's prior weight is constructed recursively.
• The means and standard deviations of the potential clusters are defined as normal and half-normal distributions.