Bayesian Conjugacy in Probit, Tobit, Multinomial Probit and Extensions: A Review and New Results

Anceschi, Niccolò; Fasano, Augusto; Durante, Daniele; Zanella, Giacomo

doi:10.1080/01621459.2023.2169150

Cited by 9 publications

(2 citation statements)

References 96 publications

(57 reference statements)

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“…At last, Traditional linear regression methods may lead to significant deviation in results when analyzing the chemical industry in Jiangsu Province, due to the limited sample of research objects. Additionally, spatial econometric methods may not be effective for achieving the estimation of sub-regional influencing factors, as the dependent variable are subject to certain constraints (Anceschi et al, 2023). For these reasons, this paper analyzed the impact of WECC on GTFP based on Tobit model.…”

Section: Introductionmentioning

confidence: 99%

The Impacts of Water Ecological Carrying Capacity on Green Total Factor Productivity: The Case of Chemical Industry in Jiangsu, China

Gu,

Chen,

et al. 2024

Sage Open

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The chemical industry is not only a crucial sector of national economy, but also a significant consumer of water resources and a major initiator of water pollution. The sustainable development of this sector is intricately linked to the regional water ecological carrying capacity (WECC). Based on SBM-DEA and Global Moran’s I, the green total factor productivity (GTFP) and spatial correlation characteristics of the chemical industry in 13 cities within China’s chemical agglomeration region in Jiangsu Province were estimated from 2015 to 2019. By combining the WECC results, the Tobit model was employed to reveal the driving factors of WECC in optimizing GTFP. The results indicated that the regional WECC in southern Jiangsu was increasing compared with that in northern Jiangsu, which promoted the growth of GTFP. WECC has been a positive radiation-driven effect since 2017, and the optimization of the various subsystems of WECC has had a different impact on GTFP. For the sustainable development of Jiangsu’s chemical industry, effective water resource policies should be formulated by the government, while enterprises need to pursue sustained structural adjustments.

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Section: Introductionmentioning

confidence: 99%

The Impacts of Water Ecological Carrying Capacity on Green Total Factor Productivity: The Case of Chemical Industry in Jiangsu, China

Gu,

Chen,

et al. 2024

Sage Open

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“…In fact, in small-to-moderate sample size settings, general posterior distributions induced by routinely-implemented statistical models, such as Bayesian generalized linear models, display a non-Gaussian behavior, mainly due to skewness (e.g., Kuss, Rasmussen and Herbrich, 2005;Challis and Barber, 2012;Durante, 2019). For example, Durante (2019), and Anceschi et al (2023) have recently proved that, under a broad class of priors which includes multivariate normals, the posterior distribution induced by probit, multinomial probit and tobit models belongs to a skewed generalization of the Gaussian dis- tribution known as unified skew-normal (SUN) (Arellano-Valle and Azzalini, 2006). More generally, available extensions of Gaussian deterministic approximations which account, either explicitly or implicitly, for skewness (e.g., Rue, Martino and Chopin, 2009;Challis and Barber, 2012;Fasano, Durante and Zanella, 2022) have shown evidence of improved empirical accuracy relative to the Gaussian counterparts.…”

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confidence: 99%

Skewed Bernstein-von Mises theorem and skew-modal approximations

Durante¹,

Pozza²,

Szabó³

2023

Preprint

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Deterministic Gaussian approximations of intractable posterior distributions are common in Bayesian inference. From an asymptotic perspective, a theoretical justification in regular parametric settings is provided by the Bernstein-von Mises theorem. However, such a limiting behavior may require a large sample size before becoming visible in practice. In fact, in situations with small-to-moderate sample size, even simple parametric models often yield posterior distributions which are far from resembling a Gaussian shape, mainly due to skewness. In this article, we provide rigorous theoretical arguments for such a behavior by deriving a novel limiting law that coincides with a closed-form and tractable skewed generalization of Gaussian densities, and yields a total variation distance from the exact posterior whose convergence rate can be shown to be of order 1/n, up to a logarithmic factor. Such a result provides a substantial accuracy improvement over the classical Bernstein-von Mises theorem whose convergence rate, under similar conditions, is of order 1/ √ n, again up to a logarithmic factor. In contrast to higher-order approximations based on, e.g., Edgeworth expansions, which require finite truncations for inference, possibly leading to even negative densities, our theory characterizes the limiting behavior of Bayesian posteriors with respect to a sequence of valid and tractable densities. These advancements further motivate a practical plug-in version which replaces the unknown model parameters with the corresponding maximum-a-posteriori estimate to obtain a novel skew-modal approximation achieving the same improved rate of convergence of our skewed Bernstein-von Mises theorem. Extensive simulations and a real-data application confirm that our new theory closely matches the empirical behavior observed in practice even in finite, possibly small, sample regimes. The proposed skew-modal approximation further exhibits improved accuracy not only relative to classical Laplace approximation, but also with respect to state-of-the-art Gaussian and non-Gaussian approximations from mean-field variational Bayes and expectation-propagation.* Co-funded by the European Union (ERC, BigBayesUQ, project number: 101041064). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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