Monitoring count data without a Phase I study: A Poisson case on industrial printers

Statistical Process Control (SPC) is traditionally based on a two-phase framework: a reference sample is required to estimate in-control parameters (Phase I) before active monitoring (Phase II) begins. However, in many contemporary industrial and service settings, such a sample is either unavailable or heavily contaminated by heterogeneity, structural changes, or sporadic anomalies. Under these conditions, classical Phase I and Phase II schemes are difficult to justify. This challenge arose while attempting to remotely monitor industrial printers, prompting a search for alternatives that led to the Bayesian outlier framework proposed by Box and Tiao. This article revisits that framework and argues that its core principles provide a coherent basis for SPC when reliable Phase I data are absent. In the Box and Tiao formulation, observations arise from a mixture of two components sharing a common mean but differing in dispersion; this allows for simultaneous parameter estimation and outlier identification through posterior probabilities. Using the industrial printer case as a primary example, this article demonstrates how this logic can be extended from the original Normal distribution to Poisson data – such as error counts – which are frequently encountered in SPC applications.