Statistics

2602 Submissions

[2] viXra:2602.0155 [pdf] submitted on 2026-02-26 09:49:23

Nested Sampling: A Critical and Comprehensive Theoretical Guide

Authors: L. Martino, F. Llorente
Comments: 28 Pages.

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, clarifying both its theoretical foundations and practical challenges. We provide a thorough description of the NS procedure, emphasizing both its strengths and potential limitations. 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.
Category: Statistics

[1] viXra:2602.0150 [pdf] submitted on 2026-02-26 21:24:53

A Unifying View of Multiple-Try Metropolis and Particle Metropolis-Hastings Algorithms

Authors: L. Martino
Comments: 27 Pages.

Markov chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) methods are cornerstone techniques for Bayesian inference and stochastic optimization. The multiple-try Metropolis (MTM) algorithm generalizes the Metropolis-Hastings (MH) scheme by selecting the next state from a set of weighted candidates, improving exploration of the state space. Particle Metropolis-Hastings (PMH) integrates MCMC and SMC ideas to efficiently tackle high-dimensional targets with sequentially factorized structures, embedding a particle filter within an MH framework. While both approaches have been extensively studied, particularly for state-space models, their relationship has not been fully explored. In this work, we examine the connections and distinctions between MTM and PMH schemes, which motivates the design of novel, highly efficient algorithms for filtering and smoothing. Among these, we introduce a particle multiple-try Metropolis (P-MTM) method, which demonstrates excellent performance across a range of numerical experiments.
Category: Statistics