  We consider the Bayesian analysis of models in which the unknown distribution
of the outcomes is specified up to a set of conditional moment restrictions.
The nonparametric exponentially tilted empirical likelihood function is
constructed to satisfy a sequence of unconditional moments based on an
increasing (in sample size) vector of approximating functions (such as tensor
splines based on the splines of each conditioning variable). For any given
sample size, results are robust to the number of expanded moments. We derive
Bernstein-von Mises theorems for the behavior of the posterior distribution
under both correct and incorrect specification of the conditional moments,
subject to growth rate conditions (slower under misspecification) on the number
of approximating functions. A large-sample theory for comparing different
conditional moment models is also developed. The central result is that the
marginal likelihood criterion selects the model that is less misspecified. We
also introduce sparsity-based model search for high-dimensional conditioning
variables, and provide efficient MCMC computations for high-dimensional
parameters. Along with clarifying examples, the framework is illustrated with
real-data applications to risk-factor determination in finance, and causal
inference under conditional ignorability.
