Reconstructing the energy landscape of a distribution from Monte Carlo samples

TL;DR

This paper introduces a method to reconstruct and analyze the energy landscape of a distribution from Monte Carlo samples, revealing detailed topological and statistical features such as local minima and barriers.

Contribution

It proposes a novel computational approach to estimate the energy landscape's hierarchical tree from samples, applicable to any distribution with connectedness, enhancing understanding of complex distributions.

Findings

Accurately reconstructs the energy landscape for multimodal distributions.

Provides detailed insights into local modes and barriers in posterior distributions.

Outperforms standard methods in Bayesian DNA sequence segmentation analysis.

Abstract

Defining the energy function as the negative logarithm of the density, we explore the energy landscape of a distribution via the tree of sublevel sets of its energy. This tree represents the hierarchy among the connected components of the sublevel sets. We propose ways to annotate the tree so that it provides information on both topological and statistical aspects of the distribution, such as the local energy minima (local modes), their local domains and volumes, and the barriers between them. We develop a computational method to estimate the tree and reconstruct the energy landscape from Monte Carlo samples simulated at a wide energy range of a distribution. This method can be applied to any arbitrary distribution on a space with defined connectedness. We test the method on multimodal distributions and posterior distributions to show that our estimated trees are accurate compared to theoretical values. When used to perform Bayesian inference of DNA sequence segmentation, this approach reveals much more information than the standard approach based on marginal posterior distributions.