Bayesian nonparametric analysis of reversible Markov chains
Sergio Bacallado, Stefano Favaro, Lorenzo Trippa
The Annals of Statistics · 2013 · Volume 41 · Issue 2
Abstract
We introduce a three-parameter random walk with reinforcement, called the $(\theta,\alpha,\beta)$ scheme, which generalizes the linearly edge reinforced random walk to uncountable spaces. The parameter $\beta$ smoothly tunes the $(\theta,\alpha,\beta)$ scheme between this edge reinforced random walk and the classical exchangeable two-parameter Hoppe urn scheme, while the parameters $\alpha$ and $\theta$ modulate how many states are typically visited. Resorting to de Finetti’s theorem for Markov chains, we use the $(\theta,\alpha,\beta)$ scheme to define a nonparametric prior for Bayesian analysis of reversible Markov chains. The prior is applied in Bayesian nonparametric inference for species sampling problems with data generated from a reversible Markov chain with an unknown transition kernel. As a real example, we analyze data from molecular dynamics simulations of protein folding.
Keywords
62C10, 62M02, Bayesian nonparametrics, mixtures of Markov chains, molecular dynamics, reinforced random walks, reversibility, Species sampling, two-parameter Hoppe urn
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