# Bayesian statistics

## Dirichlet Process Mixture Models Part III: Chinese Restaurant Process vs Stick-breaking

In the previous entry of what has evidently become a series on modelling binary mixtures with Dirichlet Processes (part 1 discussed using pymc3 and part 2 detailed writing custom Gibbs samplers), I ended by stating that I’d like to look into writing a Gibbs sampler using the stick-breaking formulation of the Dirichlet Process, in contrast to the Chinese Restaurant Process (CRP) version I’d just implemented. Actually coding this up this was rather straight forward and took less time than I expected, but I found the differences and similarities between these two same ways of expressing the same mathematical model interesting enough for a post of its own.

## Writing Gibbs Samplers for clustering binary data using Dirichlet Processes

Back at the start of the year (which really doesn’t seem like that long a time ago) I was looking at using Dirichlet Processes to cluster binary data using PyMC3. I was unable to get the PyMC3 mixture model API working using the general purpose Gibbs Sampler, but after some tweaking of a custom likelihood function I got something reasonable-looking working using Variational Inference (VI). While this was still useful for exploratory analysis purposes, I’d prefer to use MCMC sampling so that I have more confidence in the groupings (since VI only approximates the posterior) in case I wanted to use these groups to generate further research questions.

## Modelling Bernoulli Mixture Models with Dirichlet Processes in PyMC

I’ve been spending a lot of time over the last week getting Theano working on Windows playing with Dirichlet Processes for clustering binary data using PyMC3. While there is a great tutorial for mixtures of univariate distributions, there isn’t a lot out there for multivariate mixtures, and Bernoulli mixtures in particular. This notebook shows an example of fitting such a model using PyMC3 and highlights the importance of careful parameterisation as well as demonstrating that variational inference can prove advantageous over standard sampling methods like NUTS for such problems.