  Modeled along the truncated approach in Panigrahi (2016), selection-adjusted
inference in a Bayesian regime is based on a selective posterior. Such a
posterior is determined together by a generative model imposed on data and the
selection event that enforces a truncation on the assumed law. The effective
difference between the selective posterior and the usual Bayesian framework is
reflected in the use of a truncated likelihood. The normalizer of the truncated
law in the adjusted framework is the probability of the selection event; this
is typically intractable and it leads to the computational bottleneck in
sampling from such a posterior. The current work lays out a primal-dual
approach of solving an approximating optimization problem to provide valid
post-selective Bayesian inference. The selection procedures are posed as
data-queries that solve a randomized version of a convex learning program which
have the advantage of preserving more left-over information for inference. We
propose a randomization scheme under which the optimization has separable
constraints that result in a partially separable objective in lower dimensions
for many commonly used selective queries to approximate the otherwise
intractable selective posterior. We show that the approximating optimization
under a Gaussian randomization gives a valid exponential rate of decay for the
selection probability on a large deviation scale. We offer a primal-dual method
to solve the optimization problem leading to an approximate posterior; this
allows us to exploit the usual merits of a Bayesian machinery in both low and
high dimensional regimes where the underlying signal is effectively sparse. We
show that the adjusted estimates empirically demonstrate better frequentist
properties in comparison to the unadjusted estimates based on the usual
posterior, when applied to a wide range of constrained, convex data queries.
