Sequential normal predictive monitoring ($u_k=m_k+{\Phi }^{-1}(1-\alpha )\sqrt{v_k+{\sigma }^2}$)

For repeated independent $N(\theta ,{\sigma }^2)$ measurements with prior $\theta \sim N(\mu ,{\tau }^2)$ and known scales, normal-normal conjugacy gives $v_k=({\tau }^{-2}+k{\sigma }^{-2}{)}^{-1}$ and $m_k=v_k(\mu {\tau }^{-2}+{\sigma }^{-2}{\sum }_{j=1}^k{y}_j)$. The next measurement has posterior predictive distribution $N(m_k,v_k+{\sigma }^2)$. Its one-sided conditional posterior-predictive false-alarm level $\alpha$ yields the displayed limit; this is a per-measurement probability, not a simultaneous false-alarm guarantee across repeated monitoring. Predictive width tends to the irreducible measurement width, while the center adapts to the individual's data. To assess a held-out measurement, fit with earlier measurements only.