Imprecise Dirichlet process
In probability theory and statistics, the Dirichlet process (DP) is one of the most popular Bayesian nonparametric models.
It was introduced by Thomas Ferguson[1] as a prior over probability distributions.
(the concentration parameter) is a positive real number (it is often denoted as
According to the Bayesian paradigm these parameters should be chosen based on the available prior information on the domain.
To address this issue, the only prior that has been proposed so far is the limiting DP obtained for
From an a-priori point of view, the main criticism is that taking
[4] Moreover, a-posteriori, it assigns zero probability to any set that does not include the observations.
[2] The imprecise Dirichlet[5] process has been proposed to overcome these issues.
More precisely, the imprecise Dirichlet process (IDP) is defined as follows: where
In other words, the IDP is the set of all Dirichlet processes (with a fixed
with respect to the Dirichlet process is One of the most remarkable properties of the DP priors is that the posterior distribution of
is an atomic probability measure (Dirac's delta) centered at
, we can exploit the previous equations to derive prior and posterior expectations.
A way to characterize inferences for the IDP is by computing lower and upper bounds for the expectation of
From the above expressions of the lower and upper bounds, it can be observed that the range of
In other words, by specifying the IDP, we are not giving any prior information on the value of the expectation of
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
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).
determines how quickly lower and upper posterior expectations converge at the increase of the number of observations,
can also be chosen to have some desirable frequentist properties (e.g., credible intervals to be calibrated frequentist intervals, hypothesis tests to be calibrated for the Type I error, etc.
real random variables with cumulative distribution function
and the property of the Dirichlet process, it can be shown that the posterior distribution of
Dirichlet processes are frequently used in Bayesian nonparametric statistics.
The Imprecise Dirichlet Process can be employed instead of the Dirichlet processes in any application in which prior information is lacking (it is therefore important to model this state of prior ignorance).
In this respect, the Imprecise Dirichlet Process has been used for nonparametric hypothesis testing, see the Imprecise Dirichlet Process statistical package.
Based on the Imprecise Dirichlet Process, Bayesian nonparametric near-ignorance versions of the following classical nonparametric estimators have been derived: the Wilcoxon rank sum test[5] and the Wilcoxon signed-rank test.
[6] A Bayesian nonparametric near-ignorance model presents several advantages with respect to a traditional approach to hypothesis testing.
In this case, the Imprecise Dirichlet Process reduces to the Imprecise Dirichlet model proposed by Walley[7] as a model for prior (near)-ignorance for chances.
