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A measure of uncertainty regarding the interval constraint of normal mean elicited by two stages of a prior hierarchy.

Kim HJ - ScientificWorldJournal (2014)

Bottom Line: An objective measure of the uncertainty, regarding the interval constraint, accounted for by using the HSGM is proposed for the Bayesian inference.For this purpose, we drive a maximum entropy prior of the normal mean, eliciting the uncertainty regarding the interval constraint, and then obtain the uncertainty measure by considering the relationship between the maximum entropy prior and the marginal prior of the normal mean in HSGM.Bayesian estimation procedure of HSGM is developed and two numerical illustrations pertaining to the properties of the uncertainty measure are provided.

View Article: PubMed Central - PubMed

Affiliation: Department of Statistics, Dongguk University, Seoul Campus, Pil-Dong 3Ga, Chung-Gu, Seoul 100-715, Republic of Korea.

ABSTRACT
This paper considers a hierarchical screened Gaussian model (HSGM) for Bayesian inference of normal models when an interval constraint in the mean parameter space needs to be incorporated in the modeling but when such a restriction is uncertain. An objective measure of the uncertainty, regarding the interval constraint, accounted for by using the HSGM is proposed for the Bayesian inference. For this purpose, we drive a maximum entropy prior of the normal mean, eliciting the uncertainty regarding the interval constraint, and then obtain the uncertainty measure by considering the relationship between the maximum entropy prior and the marginal prior of the normal mean in HSGM. Bayesian estimation procedure of HSGM is developed and two numerical illustrations pertaining to the properties of the uncertainty measure are provided.

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Plots of (1 − α) for different values of δ ∈ (0,1) and σ2 for the cases where (a) Θ = [−1.0,0] and (b) Θ = [−1.0,1.5].
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fig2: Plots of (1 − α) for different values of δ ∈ (0,1) and σ2 for the cases where (a) Θ = [−1.0,0] and (b) Θ = [−1.0,1.5].

Mentions: Theorem 4 provides an exact measure of the uncertainty about the priori interval constraint on θ accounted for by HSGM, and it shows that the uncertainty measure is different from that of O'Hagan and Leonald [1] which mainly depends on the first stage variance, σ02, of θ in HSGM. Theorem 4 also indicates that HSGM in (2) can be used to elicit the priori uncertain interval information associated with π3(θ). Further, the entropy of the two-stage prior θ defined by the HSGM (i.e., p(θ) in (4)) can be calculated by using the formula of Ent(π3(θ)) in (23). We can visualize the degree of uncertainty about the priori interval constraint, θ ∈ [a, b], by plotting 1 − α for different values of δ ∈ (0,1) in Figure 2.


A measure of uncertainty regarding the interval constraint of normal mean elicited by two stages of a prior hierarchy.

Kim HJ - ScientificWorldJournal (2014)

Plots of (1 − α) for different values of δ ∈ (0,1) and σ2 for the cases where (a) Θ = [−1.0,0] and (b) Θ = [−1.0,1.5].
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4109128&req=5

fig2: Plots of (1 − α) for different values of δ ∈ (0,1) and σ2 for the cases where (a) Θ = [−1.0,0] and (b) Θ = [−1.0,1.5].
Mentions: Theorem 4 provides an exact measure of the uncertainty about the priori interval constraint on θ accounted for by HSGM, and it shows that the uncertainty measure is different from that of O'Hagan and Leonald [1] which mainly depends on the first stage variance, σ02, of θ in HSGM. Theorem 4 also indicates that HSGM in (2) can be used to elicit the priori uncertain interval information associated with π3(θ). Further, the entropy of the two-stage prior θ defined by the HSGM (i.e., p(θ) in (4)) can be calculated by using the formula of Ent(π3(θ)) in (23). We can visualize the degree of uncertainty about the priori interval constraint, θ ∈ [a, b], by plotting 1 − α for different values of δ ∈ (0,1) in Figure 2.

Bottom Line: An objective measure of the uncertainty, regarding the interval constraint, accounted for by using the HSGM is proposed for the Bayesian inference.For this purpose, we drive a maximum entropy prior of the normal mean, eliciting the uncertainty regarding the interval constraint, and then obtain the uncertainty measure by considering the relationship between the maximum entropy prior and the marginal prior of the normal mean in HSGM.Bayesian estimation procedure of HSGM is developed and two numerical illustrations pertaining to the properties of the uncertainty measure are provided.

View Article: PubMed Central - PubMed

Affiliation: Department of Statistics, Dongguk University, Seoul Campus, Pil-Dong 3Ga, Chung-Gu, Seoul 100-715, Republic of Korea.

ABSTRACT
This paper considers a hierarchical screened Gaussian model (HSGM) for Bayesian inference of normal models when an interval constraint in the mean parameter space needs to be incorporated in the modeling but when such a restriction is uncertain. An objective measure of the uncertainty, regarding the interval constraint, accounted for by using the HSGM is proposed for the Bayesian inference. For this purpose, we drive a maximum entropy prior of the normal mean, eliciting the uncertainty regarding the interval constraint, and then obtain the uncertainty measure by considering the relationship between the maximum entropy prior and the marginal prior of the normal mean in HSGM. Bayesian estimation procedure of HSGM is developed and two numerical illustrations pertaining to the properties of the uncertainty measure are provided.

Show MeSH