A statistical test for Nested Sampling algorithms

TL;DR

This paper introduces a statistical test called the Shrinkage Test to verify the uniformity assumption in nested sampling algorithms, revealing issues in existing methods and proposing a more robust alternative.

Contribution

The paper develops the Shrinkage Test for nested sampling, demonstrating its effectiveness in identifying algorithm failures and proposing a new robust algorithm, RADFRIENDS.

Findings

Some existing algorithms fail the Shrinkage Test due to over-optimisation.

The RADFRIENDS algorithm is robust but less efficient than MULTINEST.

The Shrinkage Test provides a controlled environment for verifying nested sampling algorithms.

Abstract

Nested sampling is an iterative integration procedure that shrinks the prior volume towards higher likelihoods by removing a "live" point at a time. A replacement point is drawn uniformly from the prior above an ever-increasing likelihood threshold. Thus, the problem of drawing from a space above a certain likelihood value arises naturally in nested sampling, making algorithms that solve this problem a key ingredient to the nested sampling framework. If the drawn points are distributed uniformly, the removal of a point shrinks the volume in a well-understood way, and the integration of nested sampling is unbiased. In this work, I develop a statistical test to check whether this is the case. This "Shrinkage Test" is useful to verify nested sampling algorithms in a controlled environment. I apply the shrinkage test to a test-problem, and show that some existing algorithms fail to pass it due to over-optimisation. I then demonstrate that a simple algorithm can be constructed which is robust against this type of problem. This RADFRIENDS algorithm is, however, inefficient in comparison to MULTINEST.