arXiv subjects

Augusto Fasano

Publications and source records attributed to Augusto Fasano.

11 recordsLinked to original sources

arXiv2026

Mean-field Variational Bayes for Sparse Probit Regression

We consider Bayesian variable selection for binary outcomes under a probit link with a spike-and-slab prior on the regression coefficients. Motivated by the computational challenges encountered by Markov chain Monte Carlo (MCMC) samplers in high-dimensional regimes, we develop a mean-field variational Bayes approximation in which all variational factors admit closed-form updates, and the evidence lower bound is available in closed form. This, in turn, allows the development of an efficient coordinate ascent variational inference algorithm to find the optimal values of the variational parameters. The approach produces posterior inclusion probabilities and parameter estimates, enabling interpretable selection and prediction within a single framework. As shown in both simulated and real data applications, the proposed method successfully identifies the important variables and is orders of magnitude faster than MCMC, while maintaining comparable accuracy.

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arXiv2024

Scalable expectation propagation for generalized linear models

Generalized linear models (GLMs) arguably represent the standard approach for statistical regression beyond the Gaussian likelihood scenario. When Bayesian formulations are employed, the general absence of a tractable posterior distribution has motivated the development of deterministic approximations, which are generally more scalable than sampling techniques. Among them, expectation propagation (EP) showed extreme accuracy, usually higher than many variational Bayes solutions. However, the higher computational cost of EP posed concerns about its practical feasibility, especially in high-dimensional settings. We address these concerns by deriving a novel efficient formulation of EP for GLMs, whose cost scales linearly in the number of covariates p. This reduces the state-of-the-art O(p^2 n) per-iteration computational cost of the EP routine for GLMs to O(p n min{p,n}), with n being the sample size. We also show that, for binary models and log-linear GLMs approximate predictive means can be obtained at no additional cost. To preserve efficient moment matching for count data, we propose employing a combination of log-normal Laplace transform approximations, avoiding numerical integration. These novel results open the possibility of employing EP in settings that were believed to be practically impossible. Improvements over state-of-the-art approaches are illustrated both for simulated and real data. The efficient EP implementation is available at https://github.com/niccoloanceschi/EPglm.

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arXiv2023

Efficient expectation propagation for posterior approximation in high-dimensional probit models

Bayesian binary regression is a prosperous area of research due to the computational challenges encountered by currently available methods either for high-dimensional settings or large datasets, or both. In the present work, we focus on the expectation propagation (EP) approximation of the posterior distribution in Bayesian probit regression under a multivariate Gaussian prior distribution. Adapting more general derivations in Anceschi et al. (2023), we show how to leverage results on the extended multivariate skew-normal distribution to derive an efficient implementation of the EP routine having a per-iteration cost that scales linearly in the number of covariates. This makes EP computationally feasible also in challenging high-dimensional settings, as shown in a detailed simulation study.

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arXiv2023

Efficient computation of predictive probabilities in probit models via expectation propagation

Binary regression models represent a popular model-based approach for binary classification. In the Bayesian framework, computational challenges in the form of the posterior distribution motivate still-ongoing fruitful research. Here, we focus on the computation of predictive probabilities in Bayesian probit models via expectation propagation (EP). Leveraging more general results in recent literature, we show that such predictive probabilities admit a closed-form expression. Improvements over state-of-the-art approaches are shown in a simulation study.

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arXiv2023

Expectation propagation for the smoothing distribution in dynamic probit

The smoothing distribution of dynamic probit models with Gaussian state dynamics was recently proved to belong to the unified skew-normal family. Although this is computationally tractable in small-to-moderate settings, it may become computationally impractical in higher dimensions. In this work, adapting a recent more general class of expectation propagation (EP) algorithms, we derive an efficient EP routine to perform inference for such a distribution. We show that the proposed approximation leads to accuracy gains over available approximate algorithms in a financial illustration.

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arXiv2023

Bayesian conjugacy in probit, tobit, multinomial probit and extensions: A review and new results

A broad class of models that routinely appear in several fields can be expressed as partially or fully discretized Gaussian linear regressions. Besides including basic Gaussian response settings, this class also encompasses probit, multinomial probit and tobit regression, among others, thereby yielding to one of the most widely-implemented families of models in applications. The relevance of such representations has stimulated decades of research in the Bayesian field, mostly motivated by the fact that, unlike for Gaussian linear regression, the posterior distribution induced by such models does not seem to belong to a known class, under the commonly-assumed Gaussian priors for the coefficients. This has motivated several solutions for posterior inference relying on sampling-based strategies or on deterministic approximations that, however, still experience computational and accuracy issues, especially in high dimensions. The scope of this article is to review, unify and extend recent advances in Bayesian inference and computation for this class of models. To address such a goal, we prove that the likelihoods induced by these formulations share a common analytical structure that implies conjugacy with a broad class of distributions, namely the unified skew-normals (SUN), that generalize Gaussians to skewed contexts. This result unifies and extends recent conjugacy properties for specific models within the class analyzed, and opens avenues for improved posterior inference, under a broader class of formulations and priors, via novel closed-form expressions, i.i.d. samplers from the exact SUN posteriors, and more accurate and scalable approximations from VB and EP. Such advantages are illustrated in simulations and are expected to facilitate the routine-use of these core Bayesian models, while providing a novel framework to study theoretical properties and develop future extensions.

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arXiv2022

Bayesian Inference for the Multinomial Probit Model under Gaussian Prior Distribution

Multinomial probit (mnp) models are fundamental and widely-applied regression models for categorical data. Fasano and Durante (2022) proved that the class of unified skew-normal distributions is conjugate to several mnp sampling models. This allows to develop Monte Carlo samplers and accurate variational methods to perform Bayesian inference. In this paper, we adapt the abovementioned results for a popular special case: the discrete-choice mnp model under zero mean and independent Gaussian priors. This allows to obtain simplified expressions for the parameters of the posterior distribution and an alternative derivation for the variational algorithm that gives a novel understanding of the fundamental results in Fasano and Durante (2022) as well as computational advantages in our special settings.

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arXiv2022

Scalable and Accurate Variational Bayes for High-Dimensional Binary Regression Models

Modern methods for Bayesian regression beyond the Gaussian response setting are often computationally impractical or inaccurate in high dimensions. In fact, as discussed in recent literature, bypassing such a trade-off is still an open problem even in routine binary regression models, and there is limited theory on the quality of variational approximations in high-dimensional settings. To address this gap, we study the approximation accuracy of routinely-used mean-field variational Bayes solutions in high-dimensional probit regression with Gaussian priors, obtaining novel and practically relevant results on the pathological behavior of such strategies in uncertainty quantification, point estimation and prediction. Motivated by these results, we further develop a new partially-factorized variational approximation for the posterior of the probit coefficients which leverages a representation with global and local variables but, unlike for classical mean-field assumptions, it avoids a fully factorized approximation, and instead assumes a factorization only for the local variables. We prove that the resulting approximation belongs to a tractable class of unified skew-normal densities that crucially incorporates skewness and, unlike for state-of-the-art mean-field solutions, converges to the exact posterior density as p goes to infinity. To solve the variational optimization problem, we derive a tractable CAVI algorithm that easily scales to p in the tens of thousands, and provably requires a number of iterations converging to 1 as p goes to infinity. Such findings are also illustrated in extensive empirical studies where our novel solution is shown to improve the approximation accuracy of mean-field variational Bayes for any n and p, with the magnitude of these gains being remarkable in those high-dimensional p>n settings where state-of-the-art methods are computationally impractical.

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arXiv2022

A Class of Conjugate Priors for Multinomial Probit Models which Includes the Multivariate Normal One

Multinomial probit models are routinely-implemented representations for learning how the class probabilities of categorical response data change with p observed predictors. Although several frequentist methods have been developed for estimation, inference and classification within such a class of models, Bayesian inference is still lagging behind. This is due to the apparent absence of a tractable class of conjugate priors, that may facilitate posterior inference on the multinomial probit coefficients. Such an issue has motivated increasing efforts toward the development of effective Markov chain Monte Carlo methods, but state-of-the-art solutions still face severe computational bottlenecks, especially in high dimensions. In this article, we show that the entire class of unified skew-normal (SUN) distributions is conjugate to several multinomial probit models. Leveraging this result and the SUN properties, we improve upon state-of-the-art solutions for posterior inference and classification both in terms of closed-form results for several functionals of interest, and also by developing novel computational methods relying either on independent and identically distributed samples from the exact posterior or on scalable and accurate variational approximations based on blocked partially-factorized representations. As illustrated in simulations and in a gastrointestinal lesions application, the magnitude of the improvements relative to current methods is particularly evident, in practice, when the focus is on high-dimensional studies.

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arXiv2021

A closed-form filter for binary time series

Non-Gaussian state-space models arise in several applications, and within this framework the binary time series setting provides a relevant example. However, unlike for Gaussian state-space models - where filtering, predictive and smoothing distributions are available in closed form - binary state-space models require approximations or sequential Monte Carlo strategies for inference and prediction. This is due to the apparent absence of conjugacy between the Gaussian states and the likelihood induced by the observation equation for the binary data. In this article we prove that the filtering, predictive and smoothing distributions in dynamic probit models with Gaussian state variables are, in fact, available and belong to a class of unified skew-normals (SUN) whose parameters can be updated recursively in time via analytical expressions. Also the key functionals of these distributions are, in principle, available, but their calculation requires the evaluation of multivariate Gaussian cumulative distribution functions. Leveraging SUN properties, we address this issue via novel Monte Carlo methods based on independent samples from the smoothing distribution, that can easily be adapted to the filtering and predictive case, thus improving state-of-the-art approximate and sequential Monte Carlo inference in small-to-moderate dimensional studies. Novel sequential Monte Carlo procedures that exploit the SUN properties are also developed to deal with online inference in high dimensions. Performance gains over competitors are outlined in a financial application.

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arXiv2021

Variational Inference for the Smoothing Distribution in Dynamic Probit Models

Recently, Fasano, Rebaudo, Durante and Petrone (2019) provided closed-form expressions for the filtering, predictive and smoothing distributions of multivariate dynamic probit models, leveraging on unified skew-normal distribution properties. This allows to develop algorithms to draw independent and identically distributed samples from such distributions, as well as sequential Monte Carlo procedures for the filtering and predictive distributions, allowing to overcome computational bottlenecks that may arise for large sample sizes. In this paper, we briefly review the above-mentioned closed-form expressions, mainly focusing on the smoothing distribution of the univariate dynamic probit. We develop a variational Bayes approach, extending the partially factorized mean-field variational approximation introduced by Fasano, Durante and Zanella (2019) for the static binary probit model to the dynamic setting. Results are shown for a financial application.

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