arXiv subjects

Luca Martino

Publications and source records attributed to Luca Martino.

At least 19 recordsLinked to original sources

arXiv2026

Nested Sampling: A Critical and Comprehensive Theoretical Guide

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. 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.

stat.CO↗

arXiv2026

A unifying view of contrastive learning, importance sampling, and bridge sampling for energy-based models

In the last decades, energy-based models (EBMs) have become an important class of probabilistic models in which a component of the likelihood is intractable and therefore cannot be evaluated explicitly. Consequently, parameter estimation in EBMs is challenging for conventional inference methods. In this work, we provide a unified framework that connects noise contrastive estimation (NCE), reverse logistic regression (RLR), multiple importance sampling (MIS), and bridge sampling within the context of EBMs. We further show that these methods are equivalent under specific conditions. This unified perspective clarifies relationships among existing methods and enables the development of new estimators, with the potential to improve statistical and computational efficiency. Furthermore, this study helps elucidate the success of NCE in terms of its flexibility and robustness, while also identifying scenarios in which its performance can be further improved. Hence, rather than being a purely descriptive review, this work offers a unifying perspective and additional methodological contributions. The MATLAB code used in the numerical experiments is also made freely available to support the reproducibility of the results.

cs.CE↗

arXiv2026

An index of effective number of variables for uncertainty and reliability analysis in model selection problems

An index of an effective number of variables (ENV) is introduced for model selection in nested models. This is the case, for instance, when we have to decide the order of a polynomial function or the number of bases in a nonlinear regression, choose the number of clusters in a clustering problem, or the number of features in a variable selection application (to name few examples). It is inspired by the idea of the maximum area under the curve (AUC). The interpretation of the ENV index is identical to the effective sample size (ESS) indices concerning a set of samples. The ENV index improves {drawbacks of} the elbow detectors described in the literature and introduces different confidence measures of the proposed solution. These novel measures can be also employed jointly with the use of different information criteria, such as the well-known AIC and BIC, or any other model selection procedures. Comparisons with classical and recent schemes are provided in different experiments involving real datasets. Related Matlab code is given.

stat.ME↗

arXiv2025

Data-driven informative priors for Bayesian inference with quasi-periodic data

Bayesian computational strategies for inference can be inefficient in approximating the posterior distribution in models that exhibit some form of periodicity. This is because the probability mass of the marginal posterior distribution of the parameter representing the period is usually highly concentrated in a very small region of the parameter space. Therefore, it is necessary to provide as much information as possible to the inference method through the parameter prior distribution. We intend to show that it is possible to construct a prior distribution from the data by fitting a Gaussian process (GP) with a periodic kernel. More specifically, we want to show that it is possible to approximate the marginal posterior distribution of the hyperparameter corresponding to the period in the kernel. Subsequently, this distribution can be used as a prior distribution for the inference method. We use an adaptive importance sampling method to approximate the posterior distribution of the hyperparameters of the GP. Then, we use the marginal posterior distribution of the hyperparameter related to the periodicity in order to construct a prior distribution for the period of the parametric model. This workflow is empirical Bayes, implemented as a modular (cut) transfer of a GP posterior for the period to the parametric model. We applied the proposed methodology to both synthetic and real data. We approximated the posterior distribution of the period of the GP kernel and then passed it forward as a posterior-as-prior with no feedback. Finally, we analyzed its impact on the marginal posterior distribution.

stat.ML↗

arXiv2025

Feature Selection via Graph Topology Inference for Soundscape Emotion Recognition

Research on soundscapes has shifted the focus of environmental acoustics from noise levels to the perception of sounds, incorporating contextual factors. Soundscape emotion recognition (SER) models perception using a set of features, with arousal and valence commonly regarded as sufficient descriptors of affect. In this work, we blend \emph{graph learning} techniques with a novel \emph{information criterion} to develop a feature selection framework for SER. Specifically, we estimate a sparse graph representation of feature relations using linear structural equation models (SEM) tailored to the widely used Emo-Soundscapes dataset. The resulting graph captures the relations between input features and the two emotional outputs. To determine the appropriate level of sparsity, we propose a novel \emph{generalized elbow detector}, which provides both a point estimate and an uncertainty interval. We conduct an extensive evaluation of our methods, including visualizations of the inferred relations. While several of our findings align with previous studies, the graph representation also reveals a strong connection between arousal and valence, challenging common SER assumptions.

eess.AS↗

arXiv2025

Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data

This work addresses the problem of graph learning from data following a Gaussian Graphical Model (GGM) with a time-varying mean. Graphical Lasso (GL), the standard method for estimating sparse precision matrices, assumes that the observed data follows a zero-mean Gaussian distribution. However, this assumption is often violated in real-world scenarios where the mean evolves over time due to external influences, trends, or regime shifts. When the mean is not properly accounted for, applying GL directly can lead to estimating a biased precision matrix, hence hindering the graph learning task. To overcome this limitation, we propose Graphical Lasso with Adaptive Targeted Adaptive Importance Sampling (GL-ATAIS), an iterative method that jointly estimates the time-varying mean and the precision matrix. Our approach integrates Bayesian inference with frequentist estimation, leveraging importance sampling to obtain an estimate of the mean while using a regularized maximum likelihood estimator to infer the precision matrix. By iteratively refining both estimates, GL-ATAIS mitigates the bias introduced by time-varying means, leading to more accurate graph recovery. Our numerical evaluation demonstrates the impact of properly accounting for time-dependent means and highlights the advantages of GL-ATAIS over standard GL in recovering the true graph structure.

cs.LG↗

arXiv2025

Optimality in importance sampling: a gentle survey

The performance of the Monte Carlo sampling methods relies on the crucial choice of a proposal density. The notion of optimality is fundamental to design suitable adaptive procedures of the proposal density within Monte Carlo schemes. This work is an exhaustive review around the concept of optimality in importance sampling. Several frameworks are described and analyzed, such as the marginal likelihood approximation for model selection, the use of multiple proposal densities, a sequence of tempered posteriors, and noisy scenarios including the applications to approximate Bayesian computation (ABC) and reinforcement learning, to name a few. Some theoretical and empirical comparisons are also provided.

stat.CO↗

arXiv2025

Effect of bismuth crystal orientations in Nernst thermomagnetic devices

In this work we report Nernst effect measurements in single crystal bismuth samples, with special emphasis on the characterization of the Nernst coefficient when the magnetic field, heat current and generated voltage are aligned along specific directions relative to the crystal axes. We found significant differences between the different orientations, reflecting the highly anisotropic electronic structure of bismuth and compatible with the Nernst characteristics obtained from polycrystalline samples. These results not only complement the experimental works published in the past but also underline the role of crystalline orientation in the context of transverse thermoelectric effects, towards an efficient design of thermomagnetic devices like the ordinary-Nernst-effect-based energy harvesters.

cond-mat.mtrl-sci↗

arXiv2023

Gradient-based Adaptive Importance Samplers

Importance sampling (IS) is a powerful Monte Carlo methodology for the approximation of intractable integrals, very often involving a target probability density function. The performance of IS heavily depends on the appropriate selection of the proposal distributions where the samples are simulated from. In this paper, we propose an adaptive importance sampler, called GRAMIS, that iteratively improves the set of proposals. The algorithm exploits geometric information of the target to adapt the location and scale parameters of those proposals. Moreover, in order to allow for a cooperative adaptation, a repulsion term is introduced that favors a coordinated exploration of the state space. This translates into a more diverse exploration and a better approximation of the target via the mixture of proposals. Moreover, we provide a theoretical justification of the repulsion term. We show the good performance of GRAMIS in two problems where the target has a challenging shape and cannot be easily approximated by a standard uni-modal proposal.

stat.CO↗

arXiv2023

CAMEO: Curiosity Augmented Metropolis for Exploratory Optimal Policies

Reinforcement Learning has drawn huge interest as a tool for solving optimal control problems. Solving a given problem (task or environment) involves converging towards an optimal policy. However, there might exist multiple optimal policies that can dramatically differ in their behaviour; for example, some may be faster than the others but at the expense of greater risk. We consider and study a distribution of optimal policies. We design a curiosity-augmented Metropolis algorithm (CAMEO), such that we can sample optimal policies, and such that these policies effectively adopt diverse behaviours, since this implies greater coverage of the different possible optimal policies. In experimental simulations we show that CAMEO indeed obtains policies that all solve classic control problems, and even in the challenging case of environments that provide sparse rewards. We further show that the different policies we sample present different risk profiles, corresponding to interesting practical applications in interpretability, and represents a first step towards learning the distribution of optimal policies itself.

cs.LG↗

arXiv2022

Polycrystalline MnBi as a transverse thermoelectric material

To assess the potential of polycrystalline MnBi as a transverse thermoelectric material, we have experimentally investigated its anomalous Nernst effect (ANE) by means of the heat flux method. We prepared MnBi samples by powder metallurgy; this technique allows the preparation of samples in arbitrary shapes with the possibility to tailor their magnetic properties. In the material exhibiting the highest remanent magnetization, we found a value of the ANE thermopower of -1.1 $μ$V/K at 1 T, after the compensation of the ordinary Nernst effect from pure bismuth present inside the polycrystalline sample. This value is comparable with those reported in the literature for single crystals.

cond-mat.mtrl-sci↗

arXiv2022

Inference over radiative transfer models using variational and expectation maximization methods

Earth observation from satellites offers the possibility to monitor our planet with unprecedented accuracy. Radiative transfer models (RTMs) encode the energy transfer through the atmosphere, and are used to model and understand the Earth system, as well as to estimate the parameters that describe the status of the Earth from satellite observations by inverse modeling. However, performing inference over such simulators is a challenging problem. RTMs are nonlinear, non-differentiable and computationally costly codes, which adds a high level of difficulty in inference. In this paper, we introduce two computational techniques to infer not only point estimates of biophysical parameters but also their joint distribution. One of them is based on a variational autoencoder approach and the second one is based on a Monte Carlo Expectation Maximization (MCEM) scheme. We compare and discuss benefits and drawbacks of each approach. We also provide numerical comparisons in synthetic simulations and the real PROSAIL model, a popular RTM that combines land vegetation leaf and canopy modeling. We analyze the performance of the two approaches for modeling and inferring the distribution of three key biophysical parameters for quantifying the terrestrial biosphere.

cs.LG↗

arXiv2022

Advances in Importance Sampling

Importance sampling (IS) is a Monte Carlo technique for the approximation of intractable distributions and integrals with respect to them. The origin of IS dates from the early 1950s. In the last decades, the rise of the Bayesian paradigm and the increase of the available computational resources have propelled the interest in this theoretically sound methodology. In this paper, we first describe the basic IS algorithm and then revisit the recent advances in this methodology. We pay particular attention to two sophisticated lines. First, we focus on multiple IS (MIS), the case where more than one proposal is available. Second, we describe adaptive IS (AIS), the generic methodology for adapting one or more proposals.

stat.CO↗

arXiv2022

Rethinking the Effective Sample Size

The effective sample size (ESS) is widely used in sample-based simulation methods for assessing the quality of a Monte Carlo approximation of a given distribution and of related integrals. In this paper, we revisit the approximation of the ESS in the specific context of importance sampling (IS). The derivation of this approximation, that we will denote as $\widehat{\text{ESS}}$, is partially available in Kong (1992). This approximation has been widely used in the last 25 years due to its simplicity as a practical rule of thumb in a wide variety of importance sampling methods. However, we show that the multiple assumptions and approximations in the derivation of $\widehat{\text{ESS}}$, makes it difficult to be considered even as a reasonable approximation of the ESS. We extend the discussion of the $\widehat{\text{ESS}}$ in the multiple importance sampling (MIS) setting, we display numerical examples, and we discuss several avenues for developing alternative metrics. This paper does not cover the use of ESS for MCMC algorithms.

stat.CO↗

arXiv2022

Optimality in Noisy Importance Sampling

In this work, we analyze the noisy importance sampling (IS), i.e., IS working with noisy evaluations of the target density. We present the general framework and derive optimal proposal densities for noisy IS estimators. The optimal proposals incorporate the information of the variance of the noisy realizations, proposing points in regions where the noise power is higher. We also compare the use of the optimal proposals with previous optimality approaches considered in a noisy IS framework.

stat.ML↗

arXiv2022

Marginal likelihood computation for model selection and hypothesis testing: an extensive review

This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratio of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing and machine learning. This article provides a comprehensive study of the state-of-the-art of the topic. We highlight limitations, benefits, connections and differences among the different techniques. Problems and possible solutions with the use of improper priors are also described. Some of the most relevant methodologies are compared through theoretical comparisons and numerical experiments.

stat.CO↗

arXiv2021

Compressed Monte Carlo with application in particle filtering

Bayesian models have become very popular over the last years in several fields such as signal processing, statistics, and machine learning. Bayesian inference requires the approximation of complicated integrals involving posterior distributions. For this purpose, Monte Carlo (MC) methods, such as Markov Chain Monte Carlo and importance sampling algorithms, are often employed. In this work, we introduce the theory and practice of a Compressed MC (C-MC) scheme to compress the statistical information contained in a set of random samples. In its basic version, C-MC is strictly related to the stratification technique, a well-known method used for variance reduction purposes. Deterministic C-MC schemes are also presented, which provide very good performance. The compression problem is strictly related to the moment matching approach applied in different filtering techniques, usually called as Gaussian quadrature rules or sigma-point methods. C-MC can be employed in a distributed Bayesian inference framework when cheap and fast communications with a central processor are required. Furthermore, C-MC is useful within particle filtering and adaptive IS algorithms, as shown by three novel schemes introduced in this work. Six numerical results confirm the benefits of the introduced schemes, outperforming the corresponding benchmark methods. A related code is also provided.

stat.CO↗

arXiv2021

Compressed particle methods for expensive models with application in Astronomy and Remote Sensing

In many inference problems, the evaluation of complex and costly models is often required. In this context, Bayesian methods have become very popular in several fields over the last years, in order to obtain parameter inversion, model selection or uncertainty quantification. Bayesian inference requires the approximation of complicated integrals involving (often costly) posterior distributions. Generally, this approximation is obtained by means of Monte Carlo (MC) methods. In order to reduce the computational cost of the corresponding technique, surrogate models (also called emulators) are often employed. Another alternative approach is the so-called Approximate Bayesian Computation (ABC) scheme. ABC does not require the evaluation of the costly model but the ability to simulate artificial data according to that model. Moreover, in ABC, the choice of a suitable distance between real and artificial data is also required. In this work, we introduce a novel approach where the expensive model is evaluated only in some well-chosen samples. The selection of these nodes is based on the so-called compressed Monte Carlo (CMC) scheme. We provide theoretical results supporting the novel algorithms and give empirical evidence of the performance of the proposed method in several numerical experiments. Two of them are real-world applications in astronomy and satellite remote sensing.

cs.CE↗