  There is a considerable amount of ongoing research on the use of Bayesian
control charts for detecting a shift from a good quality distribution to a bad
quality distribution in univariate and multivariate processes. It is widely
claimed that Bayesian control charts are economically optimal; see, for
example, Calabrese (1995) [Bayesian process control for attributes. Management
Science, DOI: 10.1287/mnsc.41.4.637] and Makis (2008) [Multivariate Bayesian
control chart. Operations Research, DOI: 10.1287/opre.1070.0495]. Some
researchers also generalize the optimality of controls defined based on
posterior probabilities to the class of partially observable Markov decision
processes. This note points out that the existing Bayesian control charts
cannot generally be optimal because many years ago an analytical counterexample
was provided by Taylor (1965) [Markovian sequential replacement processes. The
Annals of Mathematical Statistics, DOI: 10.1214/aoms/1177699796].
