Nested Sampling: A Critical and Comprehensive Theoretical Guide
Abstract
The nested sampling (NS) technique has gained widespread attention, particularly in cosmology and astronomy, due to its ability to efficiently explore high-likelihood regions - a feature akin to an implicit likelihood optimization that underlies its success. While the full theoretical derivation of NS is complex and involves several approximations, the central challenge lies in sampling from the likelihood-constrained priors, which is crucial for its performance. This work provides a comprehensive and detailed exposition of NS derivation, clarifying both its theoretical foundations and practical challenges. We provide a thorough description of the NS procedure, emphasizing both its strengths and potential limitations. The connections between NS, importance sampling (IS), and vertical Monte Carlo are also discussed in detail. In doing so, this work seeks to deepen understanding of the method and to foster the development of future enhancements, novel variants, and more efficient implementations across a wide range of scientific applications. Thus, the main contribution of this work is twofold: it serves both as a tutorial for newcomers to the field and as a critical review for experienced practitioners.
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BibTeX
@misc{martino2026nested,
title = {{Nested Sampling: A Critical and Comprehensive Theoretical Guide}},
author = {Luca Martino and Fernando Llorente},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.17916},
eprint = {2606.17916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.17916},
}
RIS
TY - UNPB
TI - Nested Sampling: A Critical and Comprehensive Theoretical Guide
AU - Luca Martino
AU - Fernando Llorente
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2606.17916
UR - https://arxiv.org/abs/2606.17916
AB - The nested sampling (NS) technique has gained widespread attention, particularly in cosmology and astronomy, due to its ability to efficiently explore high-likelihood regions - a feature akin to an implicit likelihood optimization that underlies its success. While the full theoretical derivation of NS is complex and involves several approximations, the central challenge lies in sampling from the likelihood-constrained priors, which is crucial for its performance. This work provides a comprehensive and detailed exposition of NS derivation, clarifying both its theoretical foundations and practical challenges. We provide a thorough description of the NS procedure, emphasizing both its strengths and potential limitations. The connections between NS, importance sampling (IS), and vertical Monte Carlo are also discussed in detail. In doing so, this work seeks to deepen understanding of the method and to foster the development of future enhancements, novel variants, and more efficient implementations across a wide range of scientific applications. Thus, the main contribution of this work is twofold: it serves both as a tutorial for newcomers to the field and as a critical review for experienced practitioners.
ER -