Sampling from a pseudo selective posterior using a primal-dual approach

Adopting the Bayesian methodology of adjusting for selection to provide valid inference in Panigrahi (2016), the current work proposes an approximation to a selective posterior, post randomized queries on data. Such a posterior differs from the usual one as it involves a truncated likelihood prepended with a prior belief on parameters in a Bayesian model. The truncation, imposed by selection, leads to intractability of the selective posterior, thereby posing a technical hurdle in sampling from such a posterior. We derive an optimization problem to approximate the otherwise intractable posterior, the efficiency of a sampler targeting the pseudo selective posterior depends on the computational cost involved in solving the approximating optimization problem. We adopt a primal-dual approach in the current work to obtain a reduced optimization problem that allows for scalable Bayesian inference in both low and high dimensional regimes.


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