Tolerance and prediction intervals: The Bayesian way

." Traditionally, when the data can be approximated by a Normal distribution, acceptance criteria are calculated using reference, prediction, and tolerance intervals. However, when the underlying distribution is non-normal, these methods may be unreliable, and alternative approaches are required. In the biotechnology and pharmaceutical industries, the data distribution of many quality attributes, such as viability, transduction, cluster of differentiation molecules (CD), and concentrations (titer, IFN-y), often deviate substantially from normality, and tolerance or prediction interval-related calculations that are based on normal approximations frequentist methods can yield inaccurate or even infeasible results. Bayesian methods, in contrast, provide a principled solution by accommodating a wide range of probability models tailored to the data, including those with skewness, heavy tails, or censoring. In addition, Bayesian frameworks naturally incorporate prior information, which can improve estimation precision, especially in settings with small sample sizes. In this paper, we review key Bayesian concepts and present algorithms for calculating one-sided and two-sided tolerance and prediction bounds. Through worked examples, we demonstrate the flexibility and facility of Bayesian methods in estimating tolerance and prediction limits under complex distributional settings.