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We introduce an extension of the P\'olya tree approach for constructingdistributions on the space of probability measures. By using optional stoppingand optional choice of splitting variables, the construction gives rise torandom measures that are absolutely continuous with piecewise smooth densitieson partitions that can adapt to fit the data. The resulting "optional P\'{o}lyatree" distribution has large support in total variation topology and yieldsposterior distributions that are also optional P\'{o}lya trees with computableparameter values.

Optional P\'{o}lya tree and Bayesian inference