Probing Bayesian Credible Regions Intrinsically: A Feasible Error Certification for Physical Systems.

Area of Science:

• Quantum Information Science
• Statistical Inference
• Computational Physics

Background:

• Standard Bayesian credible regions for parameter estimation face computational challenges with large datasets or high dimensions.
• Existing methods require sampling from finite parameter spaces, which becomes infeasible as the region size diminishes.
• Accurate certification of estimators like quantum states or channels is crucial but computationally intensive.

Purpose of the Study:

• To develop a novel computational framework for efficiently determining Bayesian credible region size and credibility.
• To introduce 'in-region sampling theory' as a solution for large-scale or high-dimensional parameter estimation problems.
• To provide analytical formulas for estimating credible region capacity and credibility without relying on extensive Monte Carlo sampling.

Main Methods:

• Introduction of 'in-region sampling theory' utilizing Monte Carlo methods to sample functions over the region itself.
• Definition of credible region capacity as the average l_p-norm distance (p>0) between a random region point and the estimator.
• Derivation of analytical formulas for p=2 to estimate capacity and credibility for any dimension and large datasets.

Main Results:

• Demonstrated the feasibility of computing Bayesian credible region qualities using in-region sampling, overcoming computational barriers.
• Presented analytical formulas that enable quick estimation of credible region capacity and credibility, applicable to high dimensions.
• Provided a computationally efficient alternative to traditional Bayesian certification methods, especially for quantum state tomography.

Conclusions:

• In-region sampling theory offers a significant advancement in the computational certification of Bayesian credible regions.
• The derived analytical formulas provide a rapid and accurate method for assessing region qualities without extensive sampling.
• This approach is particularly valuable in complex fields like quantum state tomography, enhancing the reliability of parameter estimation.