Posterior Distribution Charts: A Bayesian Approach for Graphically Exploring a Process Mean

Abstract

We develop a Bayesian approach for monitoring and graphically exploring a process mean and informing decisions related to process adjustment. We assume a rather general model, in which the observations are represented as a process mean plus a random error term. In contrast to previous work on Bayesian methods for monitoring a mean, we allow any Markov model for the mean. This includes a mean that wanders slowly, that is constant over periods of time with occasional random jumps or combinations thereof. The approach also allows for any distribution for the random errors, although we focus on the normal error case. We use numerical integration to update relevant posterior distributions (e.g., for the current mean or for future observations), as each new observation is obtained, in a computationally inexpensive manner. Using an example from automobile body assembly, we illustrate how the approach can inform decisions regarding whether to adjust a process. Supplementary Materials for this article, including code for implementing the charts, are available online on the journal web site.