Dynamically Determine the Number of Components in a Mixture Model

This application uses Reversible Jump Markov Chain Monte Carlo (RJMCMC) to dynamically determine the number of components in a mixture model without requiring you to specify the number of components beforehand.

Mixture Distribution Model:

The target mixture distribution is defined as a combination of K Gaussian components:

Mathematical Representation:

p(z) = Σ_k=1^K p_k · N(z | μ_k, σ_k²)

where:

• N(z | μ_k, σ_k²) is the normal density for the k-th component

• p_k represents the mixing proportion of the k-th component

• K is the number of components (inferred dynamically)

• μ_k are the component means

• σ_k are the component standard deviations

Likelihood Function:

L(data | K, {p_k, μ_k, σ_k}) = Π_i=1^N (Σ_k=1^K p_k · N(z_i | μ_k, σ_k²))

How RJMCMC Works:

Model Selection: The algorithm explores different numbers of mixture components (K) during sampling

Reversible Jumps: It can add or remove components through "birth" and "death" moves

Bayesian Inference: Uses posterior probabilities to determine the most likely number of components

Parameter Estimation: Simultaneously estimates component means, standard deviations, and mixing proportions

Key Features:

Automatic component number selection (no need to specify K beforehand)

Real-time visualization of the sampling process

Convergence diagnostics (ESS, Gelman-Rubin statistic)

Posterior distribution of mixture components