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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 DiffEnt = Ent(π3(θ)) − Ent(π2(θ)) 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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fig1: Plots of DiffEnt = Ent(π3(θ)) − Ent(π2(θ)) 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: Each graph of Figure 1 depicts the difference between Ent(π3(θ)) and Ent(π2(θ)) as a function of δ ∈ (0,1) for three values of σ2, two cases of Θ = [a, b], and μ = 0. In Figure 1, the difference is denoted by DiffEnt. Since each graph coincides with the results of Corollary 3, we can obtain the following implications from the figure. (i) As expected, we see that Ent(π1(θ)) > Ent(π3(θ)) > Ent(π2(θ)) for δ ∈ (0,1). (ii) The entropy of π3(θ) is a monotone decreasing function of δ. (iii) Each entropy of the three priors increases as σ2 becomes large. (iv) DiffEnt is closely related with degree of uncertainty (i.e,. (1 − α) × 100%) for it is a monotone decreasing function of δ ∈ (0,1) and α is a function of δ. (v) DiffEnt is a monotone increasing function of σ2 for the case where a value of δ is given.


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 DiffEnt = Ent(π3(θ)) − Ent(π2(θ)) 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

fig1: Plots of DiffEnt = Ent(π3(θ)) − Ent(π2(θ)) 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: Each graph of Figure 1 depicts the difference between Ent(π3(θ)) and Ent(π2(θ)) as a function of δ ∈ (0,1) for three values of σ2, two cases of Θ = [a, b], and μ = 0. In Figure 1, the difference is denoted by DiffEnt. Since each graph coincides with the results of Corollary 3, we can obtain the following implications from the figure. (i) As expected, we see that Ent(π1(θ)) > Ent(π3(θ)) > Ent(π2(θ)) for δ ∈ (0,1). (ii) The entropy of π3(θ) is a monotone decreasing function of δ. (iii) Each entropy of the three priors increases as σ2 becomes large. (iv) DiffEnt is closely related with degree of uncertainty (i.e,. (1 − α) × 100%) for it is a monotone decreasing function of δ ∈ (0,1) and α is a function of δ. (v) DiffEnt is a monotone increasing function of σ2 for the case where a value of δ is given.

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