Standard statistical process control treats the parameters of its control chart as known. The kernel width, the threshold, the sensitivity — all fixed from a calibration set. When a new observation falls outside the control limit, the alarm triggers. When it falls inside, the process is declared normal.
The probabilistic version treats those parameters as uncertain. Bayesian inference via MCMC estimates posterior distributions for the kernel parameters rather than point values. The result is a credible band around the monitoring statistic rather than a fixed line. And this band does something useful: it narrows during normal operations and widens during faults.
The widening under faults reflects amplified epistemic uncertainty. The kernel parameters were estimated from in-control data. When the process enters an abnormal regime that the training data didn't cover, the model correctly reports less confidence in its own statistic. The control chart becomes uncertain about itself precisely when the process is uncertain.
Deterministic approaches miss this. A fixed control limit is equally confident during normal operations and during a novel fault. It triggers an alarm at the same threshold whether the violation is one standard deviation into uncharted territory or ten. The probability-based approach scales its own authority with its evidence.
The practical implication for manufacturing: a wider credible band during a fault means the operator should investigate even if the monitoring statistic hasn't crossed the threshold. The uncertainty itself is the signal.