Imprecise Dirichlet process information

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.

A Dirichlet process $\mathrm {DP} \left(s,G_{0}\right)$ is completely defined by its parameters: $G_{0}$ (the base distribution or base measure) is an arbitrary distribution and $s$ (the concentration parameter) is a positive real number (it is often denoted as $\alpha$ ). 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 $\left(s,G_{0}\right)$ of the DP, in particular the infinite dimensional one $G_{0}$ , 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 $s\rightarrow 0$ , which has been introduced under the name of Bayesian bootstrap by Rubin;^[2] in fact it can be proven that the Bayesian bootstrap is asymptotically equivalent to the frequentist bootstrap introduced by Bradley Efron.^[3] The limiting Dirichlet process $s\rightarrow 0$ has been criticized on diverse grounds. From an a-priori point of view, the main criticism is that taking $s\rightarrow 0$ is far from leading to a noninformative prior.^[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. The basic idea is to fix $s>0$ but do not choose any precise base measure $G_{0}$ .

More precisely, the imprecise Dirichlet process (IDP) is defined as follows:

~~\mathrm {IDP} :~\left\{\mathrm {DP} \left(s,G_{0}\right):~~G_{0}\in \mathbb {P} \right\}

where $\mathbb {P}$ is the set of all probability measures. In other words, the IDP is the set of all Dirichlet processes (with a fixed $s>0$ ) obtained by letting the base measure $G_{0}$ to span the set of all probability measures.

^ Ferguson, Thomas (1973). "Bayesian analysis of some nonparametric problems". Annals of Statistics. 1 (2): 209–230. doi:10.1214/aos/1176342360. MR 0350949.
^ ^a ^b Rubin D (1981). The Bayesian bootstrap. Ann. Stat. 9 130–134
^ Efron B (1979). Bootstrap methods: Another look at the jackknife. Ann. Stat. 7 1–26
^ Sethuraman, J.; Tiwari, R. C. (1981). "Convergence of Dirichlet measures and the interpretation of their parameter". Defense Technical Information Center.
^ Benavoli, Alessio; Mangili, Francesca; Ruggeri, Fabrizio; Zaffalon, Marco (2014). "Imprecise Dirichlet Process with application to the hypothesis test on the probability that X< Y". arXiv:1402.2755 [math.ST].

[1] Ferguson, Thomas (1973). "Bayesian analysis of some nonparametric problems". Annals of Statistics. 1 (2): 209–230. doi:10.1214/aos/1176342360. MR 0350949.

[Rubin1981-2] Rubin D (1981). The Bayesian bootstrap. Ann. Stat. 9 130–134

[Efron1979-3] Efron B (1979). Bootstrap methods: Another look at the jackknife. Ann. Stat. 7 1–26

[4] Sethuraman, J.; Tiwari, R. C. (1981). "Convergence of Dirichlet measures and the interpretation of their parameter". Defense Technical Information Center.

[Benavoliarxiv-5] Benavoli, Alessio; Mangili, Francesca; Ruggeri, Fabrizio; Zaffalon, Marco (2014). "Imprecise Dirichlet Process with application to the hypothesis test on the probability that X< Y". arXiv:1402.2755 [math.ST].

Imprecise Dirichlet process information

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