  Models with intractable normalizing functions have numerous applications.
Because the normalizing constants are functions of the parameters of interest,
standard Markov chain Monte Carlo cannot be used for Bayesian inference for
these models. A number of algorithms have been developed for such models. Some
have the posterior distribution as their asymptotic distribution. Other
``asymptotically inexact'' algorithms do not possess this property. There is
limited guidance for evaluating approximations based on these algorithms. Hence
it is very hard to tune them. We propose two new diagnostics that address these
problems for intractable normalizing function models. Our first diagnostic,
inspired by the second Bartlett identity, is in principle broadly applicable to
Monte Carlo approximations beyond the normalizing function problem. We develop
an approximate version of this diagnostic that is applicable to intractable
normalizing function problems. Our second diagnostic is a Monte Carlo
approximation to a kernel Stein discrepancy-based diagnostic introduced by
Gorham and Mackey (2017). We provide theoretical justification for our methods
and apply them to several algorithms in challenging simulated and real data
examples including an Ising model, an exponential random graph model, and a
Conway--Maxwell--Poisson regression model, obtaining interesting insights about
the algorithms in these contexts.
