B1271
Title: Quantile importance sampling
Authors: Jyotishka Datta - Virginia Polytechnic Institute and State University (United States) [presenting]
Nicholas Polson - University of Chicago (United States)
Abstract: In Bayesian inference, the approximation of integrals of the form $\psi = \mathbb{E}_{F}{l(X)} = \int_{\chi} l(\mathbf{x}) dF(\mathbf{x})$ is a fundamental challenge. Such integrals are crucial for evidence estimation, which is important for various purposes, including model selection and numerical analysis. The existing strategies for evidence estimation are classified into four categories: deterministic approximation, density estimation, importance sampling, and vertical representation. It is argued that the Riemann sum estimator can be used in nested sampling to achieve a $O(n^{-4})$ convergence rate faster than the usual Ergodic Central Limit Theorem, under certain regularity conditions. A brief overview of the literature on the Riemann sum estimators, the nested sampling algorithm, and its connections to vertical likelihood Monte Carlo are provided. Further theoretical and numerical arguments show how merging these two ideas may result in improved and more robust estimators for evidence estimation, especially in higher dimensional spaces. The idea of simulating the Lorenz curve that avoids the problem of intractable $\Lambda$ functions is also discussed, which is essential for vertical representation and nested sampling.
