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Bayesian nonparametric analysis of reversible Markov chains. (English) Zbl 1360.62481
Summary: 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.

MSC:
62M99 Inference from stochastic processes
62M02 Markov processes: hypothesis testing
62C10 Bayesian problems; characterization of Bayes procedures
60G50 Sums of independent random variables; random walks
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