Bayesian mixture model for univariate continuous random variable

Question

I'm quite new to the mixture models and I hope you'll help me to understand how they work.

Suppose I have a univariate continuous random variable x which represents time of a visit, and suppose that x is multimodal, i.e. it is described by more than one probability distribution. I want to find how many subpopulations are within the overall population of x.

I know that I can use finite mixture models, where I need to specify a number of components and probability distribution of each component, such as Gaussians. However, what I would like is to learn a number of components from the data and to not make assumptions regarding distribution of each component, but ideally infer it from the model. I know that Bayesian mixture model can do that, however I'm struggling to understand how the model works. In particular, how Bayesian nonparametric mixture model can be applied to continuous random variable? Is there any assumption about distribution of each component and how I should choose priors?

Answer 1

A Bayesian nonparametric mixture model could look like:

$X_i \sim N(\phi_i)$

$\phi_i \sim G$

$G \sim DP(G_0, \alpha)$

Where $\phi_i = (\mu_i, \sigma^2_i)$. DP is the Dirichlet process prior, so that $G$ is a discrete probability distribution and there can be some positive probability that $\phi_i = \phi_j, i\ne j$. $G_0$ is a Normal $\times $Inverse Chi-square which functions as a conjugate prior for the mean and variance of the components of the mixture model.

Once you can sample from the model (see Neal 2000), you need to decide whether you're actually interested in how many components there are or whether density estimation fits your needs. MCMC for such a model may use a different number of mixture components at each step. Also, the model uses circular Gaussian components to build up more complex shapes, so there's no guarantee that the clustering at any one step is good on its own.