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We present a survey of some of our recent results on Bayesian nonparametricinference for a multitude of stochastic processes. The common feature is thatthe prior distribution in the cases considered is on suitable sets of piecewiseconstant or piecewise linear functions, that differ for the specific situationsat hand. Posterior consistency and in most cases contraction rates for theestimators are presented. Numerical studies on simulated and real dataaccompany the theoretical results.

Nonparametric Bayesian inference for stochastic processes with piecewise constant priors