  The missing data issue often complicates the task of estimating generalized
linear models (GLMs). We describe why the pseudo-marginal Metropolis-Hastings
algorithm, used in this setting, is an effective strategy for parameter
estimation. This approach requires fewer assumptions, it provides joint
inferences on the parameters in the likelihood, the covariate model, and the
parameters of the missingness-mechanism, and there is no logical inconsistency
of assuming that there are multiple posterior distributions. Moreover, this
approach is asymptotically exact, just like most other Markov chain Monte Carlo
techniques. We discuss computing strategies, conduct a simulation study
demonstrating how standard errors change as a function of percent missingness,
and we use our approach on a "real-world" data set to describe how a collection
of variables influences the car crash outcomes.
