  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.
