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In problems with large amounts of missing data one must model two distinctdata generating processes: the outcome process which generates the response andthe missing data mechanism which determines the data we observe. Under theignorability condition of Rubin (1976), however, likelihood-based inference forthe outcome process does not depend on the missing data mechanism so that onlythe former needs to be estimated; partially because of this simplification,ignorability is often used as a baseline assumption. We study the implicationsof Bayesian ignorability in the presence of high-dimensional nuisanceparameters and argue that ignorability is typically incompatible with sensibleprior beliefs about the amount of selection bias. We show that, for manyproblems, ignorability directly implies that the prior on the selection bias istightly concentrated around zero. This is demonstrated on several models ofpractical interest, and the effect of ignorability on the posteriordistribution is characterized for high-dimensional linear models with a ridgeregression prior. We then show both how to build high-dimensional models whichencode sensible beliefs about the selection bias and also show that undercertain narrow circumstances ignorability is less problematic.

In Nonparametric and High-Dimensional Models, Bayesian Ignorability is an Informative Prior