The Bayesian β-binomial model to improve the inference based on the results of sampling plans: the case study of the biotoxin monitoring plan of <i>Chamelea gallina</i> natural beds

Sampling is a cornerstone of food safety monitoring, yet traditional frequentist approaches can yield unreliable inferences in the presence of rare or absent events, particularly when maximum likelihood estimates lie on the boundary of the parameter space and asymptotic approximations are not justified due to insufficient event counts. This study introduces the Bayesian β-binomial model as an effective framework to improve statistical inference in sampling plans, integrating prior knowledge with observed data to yield robust uncertainty estimates of contamination prevalence. The model was applied to ten years (2015-2024) of monitoring data on algal biotoxins in Chamelea gallina striped clams from classified harvesting areas in the Marche Region, Italy. Given the scarcity of historical data, three prior scenarios were tested-optimistic [β(1,10)], non-informative [β(1,1)], and pessimistic [β(5,10)]-to evaluate sensitivity to prior assumptions. Analytical determinations for five toxin groups (domoic acid, saxitoxin, azaspiracid, okadaic acid, yessotoxin) showed that all results, except one sample below the maximum legal limit, were under quantification thresholds. Bayesian posterior estimates confirmed a very low probability of biotoxin accumulation above limits, even under pessimistic assumptions. The findings demonstrate the model's capacity to generate stable, interpretable estimates and credible intervals, particularly valuable when the prevalence is not exactly zero but may be extremely small. Overall, the Bayesian β-binomial approach strengthens evidence-based decision-making in food safety surveillance, providing a transparent and adaptable tool for risk assessment and regulatory management.