  This work studies the large sample properties of the posterior-based
inference in the curved exponential family under increasing dimension. The
curved structure arises from the imposition of various restrictions on the
model, such as moment restrictions, and plays a fundamental role in
econometrics and others branches of data analysis. We establish conditions
under which the posterior distribution is approximately normal, which in turn
implies various good properties of estimation and inference procedures based on
the posterior. In the process we also revisit and improve upon previous results
for the exponential family under increasing dimension by making use of
concentration of measure. We also discuss a variety of applications to
high-dimensional versions of the classical econometric models including the
multinomial model with moment restrictions, seemingly unrelated regression
equations, and single structural equation models. In our analysis, both the
parameter dimension and the number of moments are increasing with the sample
size.
