[Paper Review] Nested Sampling And Likelihood Plateaus

Doris Schittenhelm, Philipp Wacker|arXiv (Cornell University)|May 18, 2020

Gaussian Processes and Bayesian Inference29 references4 citations

TL;DR

This paper rigorously justifies nested sampling for computing Bayesian evidence even when the likelihood function has non-negligible plateaus—regions where the likelihood is constant over positive prior measure. Although standard nested sampling assumes uniform sampling in the unit interval, the authors prove the integral transformation remains valid despite this assumption being violated on plateaus, and propose a modified algorithm to improve performance in such cases.

ABSTRACT

The main idea of nested sampling is to substitute the high-dimensional likelihood integral over the parameter space $Ω$ by an integral over the unit line $[0,1]$ by employing a push-forward with respect to a suitable transformation. For this substitution, it is often implicitly or explicitly assumed that samples from the prior are uniformly distributed along this unit line after having been mapped by this transformation. We show that this assumption is wrong, especially in the case of a likelihood function with plateaus. Nevertheless, we show that the substitution enacted by nested sampling works because of more interesting reasons which we lay out. Although this means that analytically, nested sampling can deal with plateaus in the likelihood function, the actual performance of the algorithm suffers under such a setting and the method fails to approximate the evidence, mean and variance appropriately. We suggest a robust implementation of nested sampling by a simple decomposition idea which demonstrably overcomes this issue.

Motivation & Objective

• To address the theoretical validity of nested sampling when the likelihood function has non-negligible plateaus.
• To challenge the implicit assumption in nested sampling that prior samples are uniformly distributed on [0,1] after transformation.
• To provide a rigorous mathematical foundation for nested sampling beyond the original assumptions, particularly in the presence of discontinuities in the survival function.
• To improve the practical performance of nested sampling in plateaued likelihood scenarios through a modified algorithmic approach.

Proposed method

• The authors analyze the push-forward measure induced by nested sampling and show that the integral transformation remains valid even when the likelihood has plateaus.
• They define problematic points as those where the likelihood is constant over a set of positive prior measure, forming a set P with μ(P) = 0 under the push-forward measure.
• Using measure-theoretic tools, they prove that the union of intervals corresponding to plateaus has zero measure under the push-forward likelihood measure.
• They establish that the uncountable union of intervals associated with plateaued likelihood values reduces to a countable union of disjoint intervals via properties of open sets in ℝ.
• A modified nested sampling algorithm is proposed that better handles plateaued likelihoods by adjusting the sampling strategy in such regions.
• Theoretical justification relies on continuity from below of measures and the structure of intervals in ℝ, particularly for half-open and open intervals.

Experimental results

Research questions

• RQ1 Is the standard nested sampling integral transformation valid when the likelihood function has a plateau of non-zero prior measure?
• RQ2 Does the implicit assumption of uniform sampling in the [0,1] interval hold when the likelihood has plateaus?
• RQ3 Can the push-forward measure in nested sampling still yield the correct evidence integral despite discontinuities in the survival function?
• RQ4 What modifications to nested sampling improve its performance in the presence of likelihood plateaus?

Key findings

• The standard nested sampling transformation remains mathematically valid even when the likelihood has non-negligible plateaus, despite the violation of the uniform sampling assumption.
• The set of points with plateaued likelihood values has prior measure zero under the push-forward measure, ensuring the integral substitution remains correct.
• The union of intervals corresponding to plateaued likelihood values can be reduced to a countable union of disjoint intervals, preserving measure-theoretic consistency.
• The push-forward measure of the likelihood over plateau intervals has zero total measure, which justifies the correctness of the transformation.
• A modified nested sampling algorithm is proposed that improves performance in plateaued likelihood scenarios by adjusting sampling dynamics.
• The theoretical foundation of nested sampling is extended to include likelihood functions with discontinuous survival functions, broadening its applicability.