page_title: Imprecise Dirichlet process
section_title: Inferences with the Imprecise Dirichlet Process
hierarchical_section_title: Imprecise Dirichlet process / Inferences with the Imprecise Dirichlet Process
caption_attribution_description: English: Lower (red) and Upper (blue) cumulative distribution function
context_page_description: In probability theory and statistics, the Dirichlet process is one of the most popular Bayesian nonparametric models. It was
  introduced by Thomas Ferguson as a prior over probability distributions.
  A Dirichlet process is completely defined by its parameters: is an arbitrary distribution and 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 of the DP, in particular the infinite dimensional one, 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, which has been introduced under
  the 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 has been criticized on diverse grounds. From an a-priori point of view, the main
  criticism is that taking is far from leading to a noninformative prior.
  Moreover, a-posteriori, it assigns zero probability to any set that does not include the observations.
context_section_description: Let  a probability distribution on  (here  is a standard Borel space with Borel -field ) and assume that .
  Then consider  a real-valued bounded function  defined on .  It is well known that the expectation of  with respect to the Dirichlet process  is
  One of the most remarkable properties of the DP priors is that the posterior distribution of  is again a DP.
  Let  be an independent and identically distributed sample from  and , then the posterior distribution of  given the observations is
  where  is an atomic probability measure (Dirac's delta) centered at . Hence,  it follows
  that 
  Therefore, for any fixed , we can exploit the previous equations to derive prior and posterior expectations.
  In the IDP  can span the set of all distributions . This implies that we will get a different prior and posterior expectation of  for any choice of  . A way to characterize inferences for the IDP is by computing lower and upper bounds for the expectation of  w.r.t. .
  A-priori these bounds are:
  the lower (upper) bound is obtained by a probability measure that puts all the mass on the infimum (supremum) of , i.e.,  with  (or respectively with ). From the above expressions of the lower and upper bounds, it can be observed that the range of   under the IDP is the same as the original range of .  In other words, by specifying the IDP, we are not giving any prior information on the value of the expectation of . A-priori, IDP is therefore a model of prior (near)-ignorance for .
  A-posteriori, IDP can learn from data. The posterior lower and upper bounds for the expectation of  are in fact given by:
  It can be observed that the posterior inferences do not depend on  .   To define the IDP, the modeler has only to choose  (the concentration parameter). This explains the meaning of the adjective near in prior near-ignorance,  because the IDP requires by the modeller the elicitation of a parameter. However, this is a simple elicitation problem for a nonparametric prior, since we only have  to choose the value of a positive scalar (there are not infinitely many parameters left in the IDP model).
  Finally, observe that for , IDP satisfies
  where . In other words, the IDP is consistent.
caption: Lower (red) and Upper (blue) cumulative distribution for the observations {−1.17, 0.44, 1.17, 3.28, 1.44, 1.98}
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