[Paper Review] On Bayesian credible sets in restricted parameter space problems and lower bounds for frequentist coverage

Éric Marchand, William E. Strawderman|arXiv (Cornell University)|Jul 31, 2012

Probabilistic and Robust Engineering Design11 references3 citations

TL;DR

This paper introduces a unified framework for constructing Bayesian credible sets in restricted parameter spaces where the parameter of interest is lower-bounded. By leveraging a spending function approach, it derives a class of credible sets with guaranteed frequentist coverage exceeding $\frac{1-\alpha}{1+\alpha}$, including an equal-tails modification of the HPD interval that matches the HPD in symmetric cases and extends coverage guarantees to non-symmetric settings where prior results were lacking.

ABSTRACT

For estimating a lower bounded parametric function in the framework of Marchand and Strawderman (2006), we provide through a unified approach a class of Bayesian confidence intervals with credibility $1-α$ and frequentist coverage probability bounded below by $\frac{1-α}{1+α}$. In cases where the underlying pivotal distribution is symmetric, the findings represent extensions with respect to the specification of the credible set achieved through the choice of a {\it spending function}, and include Marchand and Strawderman's HPD procedure result. For non-symmetric cases, the determination of a such a class of Bayesian credible sets fills a gap in the literature and includes an "equal-tails" modification of the HPD procedure. Several examples are presented demonstrating wide applicability.

Motivation & Objective

• To address the lack of analytical lower bounds for frequentist coverage in Bayesian credible sets under non-symmetric, restricted parameter spaces.
• To extend Marchand and Strawderman’s (2006) coverage result beyond symmetric pivot distributions to general location-scale and exponential family models.
• To provide a class of Bayesian credible sets—rather than a single procedure—that ensure minimal frequentist coverage exceeding $\frac{1-\alpha}{1+\alpha}$.
• To unify and generalize existing results on HPD intervals and equal-tailed intervals under parameter restrictions.
• To demonstrate the practical relevance of the spending function in shaping the frequentist performance of Bayesian intervals.

Proposed method

• The method employs a spending function interpretation to parameterize the construction of Bayesian credible sets, allowing flexible control over tail probabilities.
• It assumes the existence of a pivotal statistic $T(X,\theta)$ with a known distribution $G$ independent of $\theta$, enabling the derivation of coverage properties.
• The credible sets are defined via a prior that truncates the non-informative prior to the restricted parameter space $\theta \geq 0$, ensuring proper posterior inference.
• The approach constructs credible sets by allocating tail probabilities via a spending function, which generalizes the HPD and equal-tailed intervals.
• The frequentist coverage is analyzed by integrating over the sampling distribution of the pivotal statistic, leading to a lower bound on coverage.
• The framework applies to a wide range of models, including location-scale families, linear models, and multivariate scale families with constraints.

Experimental results

Research questions

• RQ1 Can a unified method be developed to ensure minimal frequentist coverage for Bayesian credible sets in restricted parameter spaces beyond symmetric pivot distributions?
• RQ2 What is the minimal frequentist coverage achievable by Bayesian credible sets when the underlying pivot is non-symmetric and the parameter is lower-bounded?
• RQ3 How does the choice of spending function affect the frequentist coverage of Bayesian credible sets in restricted models?
• RQ4 Can an equal-tailed credible set be constructed that matches the HPD interval in symmetric cases but maintains strong coverage in non-symmetric cases?
• RQ5 To what extent can the spending function framework generalize existing results on HPD intervals with guaranteed coverage in restricted parameter spaces?

Key findings

• The proposed class of Bayesian credible sets achieves a minimal frequentist coverage probability bounded below by $\frac{1-\alpha}{1+\alpha}$ for all $\theta \geq 0$, regardless of the symmetry of the pivot distribution.
• In symmetric cases, the equal-tails credible set coincides with the HPD interval, preserving the known coverage guarantee.
• For non-symmetric models such as the Gamma distribution with $\theta \geq 1$, the method provides the first analytical lower bound on frequentist coverage, filling a key gap in the literature.
• The spending function approach allows for a flexible class of credible sets, not limited to HPD or equal-tails, each with guaranteed coverage.
• The framework applies broadly to location-scale families, linear models with restricted linear combinations, and scale ratio estimation.
• The results demonstrate that the choice of bounds or spending function critically influences frequentist performance, precluding a universal assessment of Bayesian credible sets.