Optimizing inspection intensity under the graded approach: a Bayesian decision-theoretic model for radiation protection regulation.

The graded approach is a central principle of radiation protection regulation: regulatory requirements and oversight activities should be commensurate with the radiation risks associated with practices involving radiation sources. Although the principle is widely accepted, its practical implementation often relies on qualitative categories, expert judgement and institutional routines rather than an explicit decision model. This paper develops a Bayesian decision-theoretic formulation of inspection intensity under the graded approach. Regulated practices are represented by uncertain latent safety or compliance states, and inspection findings are treated as imperfect observations that update the regulator's beliefs. Regulatory responses are selected by minimizing expected loss, while inspection intensity is chosen by comparing the expected value of information with the resource cost and burden of inspection. The framework therefore interprets proportionality not simply as more inspection for higher-risk practices, but as inspection when the expected improvement in regulatory decision making justifies the cost. A dynamic extension is introduced in which beliefs, inspection findings, regulatory responses and underlying safety states evolve over time. An illustrative Monte Carlo simulation compares four strategies: fixed periodic inspection, risk-category inspection, uncertainty-triggered inspection and Bayesian value-of-information inspection. A sensitivity analysis varies the relative cost of limited and full inspections. The results show how a value-of-information rule can reduce cumulative expected loss relative to mechanical inspection rules under the assumed parameter values, while also adapting inspection effort as inspection costs change. The simulation is not calibrated to empirical regulatory data; its purpose is to demonstrate the logic and policy interpretation of the proposed framework. The model provides a transparent formal language for discussing proportionality, uncertainty, learning and resource allocation in radiation protection inspection programmes.