  In problems with large amounts of missing data one must model two distinct
data generating processes: the outcome process which generates the response and
the missing data mechanism which determines the data we observe. Under the
ignorability condition of Rubin (1976), however, likelihood-based inference for
the outcome process does not depend on the missing data mechanism so that only
the former needs to be estimated; partially because of this simplification,
ignorability is often used as a baseline assumption. We study the implications
of Bayesian ignorability in the presence of high-dimensional nuisance
parameters and argue that ignorability is typically incompatible with sensible
prior beliefs about the amount of selection bias. We show that, for many
problems, ignorability directly implies that the prior on the selection bias is
tightly concentrated around zero. This is demonstrated on several models of
practical interest, and the effect of ignorability on the posterior
distribution is characterized for high-dimensional linear models with a ridge
regression prior. We then show both how to build high-dimensional models which
encode sensible beliefs about the selection bias and also show that under
certain narrow circumstances ignorability is less problematic.
