Computing Entropies With Nested Sampling

TL;DR

This paper presents a novel nested sampling method to compute Shannon entropy and related measures for probability distributions that are only sampleable, demonstrated through three diverse examples.

Contribution

It introduces a new computational approach using nested sampling to evaluate entropies for distributions that cannot be directly evaluated.

Findings

Successfully applied to Gaussian, experimental design, and heavy-tailed scenarios.

Provides accurate entropy estimates where traditional methods struggle.

Enhances the toolkit for uncertainty quantification in complex models.

Abstract

The Shannon entropy, and related quantities such as mutual information, can be used to quantify uncertainty and relevance. However, in practice, it can be difficult to compute these quantities for arbitrary probability distributions, particularly if the probability mass functions or densities cannot be evaluated. This paper introduces a computational approach, based on Nested Sampling, to evaluate entropies of probability distributions that can only be sampled. I demonstrate the method on three examples: a simple gaussian example where the key quantities are available analytically; (ii) an experimental design example about scheduling observations in order to measure the period of an oscillating signal; and (iii) predicting the future from the past in a heavy-tailed scenario.