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In probability theory and statistics, the Dirichlet process is one of the most popular Bayesian nonparametric models. It wasintroduced by Thomas Ferguson as a prior over probability distributions.
A Dirichlet process D P {\displaystyle \mathrm {DP} \left} is completely defined by its parameters: G 0 {\displaystyle G_{0}} is an arbitrary distribution and s {\displaystyle s} is a positive real number.According to the Bayesian paradigm these parameters should be chosen based on the available prior information on the domain.
The question is: how should we choose the prior parameters {\displaystyle \left} of the DP, in particular the infinite dimensional one G 0 {\displaystyle G_{0}} , in case of lack of prior information?
To address this issue, the only prior that has been proposed so far is the limiting DP obtained for s → 0 {\displaystyle s\rightarrow 0} , which has been introduced underthe name of Bayesian bootstrap by Rubin; in fact it can be proven that the Bayesian bootstrap is asymptotically equivalent to the frequentist bootstrap introduced by Bradley Efron.The limiting Dirichlet process s → 0 {\displaystyle s\rightarrow 0} has been criticized on diverse grounds. From an a-priori point of view, the maincriticism is that taking s → 0 {\displaystyle s\rightarrow 0} is far from leading to a noninformative prior.Moreover, a-posteriori, it assigns zero probability to any set that does not include the observations.