  A recent paper presents the "false confidence theorem" (FCT) which has
potentially broad implications for statistical inference using Bayesian
posterior uncertainty. This theorem says that with arbitrarily large
(sampling/frequentist) probability, there exists a set which does \textit{not}
contain the true parameter value, but which has arbitrarily large posterior
probability. Since the use of Bayesian methods has become increasingly popular
in applications of science, engineering, and business, it is critically
important to understand when Bayesian procedures lead to problematic
statistical inferences or interpretations. In this paper, we consider a number
of examples demonstrating the paradoxical nature of false confidence to begin
to understand the contexts in which the FCT does (and does not) play a
meaningful role in statistical inference. Our examples illustrate that models
involving marginalization to non-linear, not one-to-one functions of multiple
parameters play a key role in more extreme manifestations of false confidence.
