The horseshoe estimator for sparse signals

Carvalho, Carlos M.; Polson, Nicholas G.; Scott, James G.

doi:10.1093/biomet/asq017

Cited by 1,207 publications

(1,467 citation statements)

References 18 publications

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“…Griffin and Brown (2011) Horseshoe Not analytically tractable Carvalho et al (2010) Discrete normal mixture p(…”

Section: Shrinkage Priormentioning

confidence: 99%

“…Due to these advantages, Bayesian penalization is becoming increasingly popular in the literature (see e.g., Alhamzawi et al, 2012;Andersen et al, 2017;Armagan et al, 2013;Bae and Mallick, 2004;Bhadra et al, 2016;Bhattacharya et al, 2012;Bornn et al, 2010;Caron and Doucet, 2008;Carvalho et al, 2010;Feng et al, 2017;Griffin and Brown, 2017;Hans, 2009;Ishwaran and Rao, 2005;Lu et al, 2016;Peltola et al, 2014;Polson and Scott, 2011;Roy and Chakraborty, 2016;Zhao et al, 2016). An active area of research investigates theoretical properties of priors for Bayesian penalization, such as the Bayesian lasso prior (for a recent overview, see Bhadra et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

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Shrinkage priors for Bayesian penalized regression.

Erp¹,

Oberski²,

Mulder³

2018

Preprint

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In linear regression problems with many predictors, penalized regression techniques are often used to guard against overfitting and to select variables relevant for predicting an outcome variable. Recently, Bayesian penalization is becoming increasingly popular in which the prior distribution performs a function similar to that of the penalty term in classical penalization. Specifically, the so-called shrinkage priors in Bayesian penalization aim to shrink small effects to zero while maintaining true large effects. Compared to classical penalization techniques, Bayesian penalization techniques perform similarly or sometimes even better, and they offer additional advantages such as readily available uncertainty estimates, automatic estimation of the penalty parameter, and more flexibility in terms of penalties that can be considered. However, many different shrinkage priors exist and the available, often quite technical, literature primarily focuses on presenting one shrinkage prior and often provides comparisons with only one or two other shrinkage priors. This can make it difficult for researchers to navigate through the many prior options and choose a shrinkage prior for the problem at hand. Therefore, the aim of this paper is to provide a comprehensive overview of the literature on Bayesian penalization. We provide a theoretical and conceptual comparison of nine different shrinkage priors and parametrize the priors, if possible, in terms of scale mixture of normal distributions to facilitate comparisons. We illustrate different characteristics and behaviors of the shrinkage priors and compare their performance in terms of prediction and variable selection in a simulation study. Additionally, we provide two empirical examples to illustrate the application of Bayesian penalization. Finally, an R package bayesreg is available online (https://github.com/sara-vanerp/bayesreg) which allows researchers to perform Bayesian penalized regression with novel shrinkage priors in an easy manner.

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“…Griffin and Brown (2011) Horseshoe Not analytically tractable Carvalho et al (2010) Discrete normal mixture p(…”

Section: Shrinkage Priormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Shrinkage priors for Bayesian penalized regression.

Erp¹,

Oberski²,

Mulder³

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…For instance, it encodes constraints such as Y 2 ⊥ ⊥ Y 4 If X is a latent variable which we are not interested in estimating, there might be no need to model explicitly its relationship to the observed variables {Y 1 ,Y 2 ,Y 3 ,Y 4 } -a task which would require extra and perhaps undesirable assumptions.…”

Section: Acyclic Directed Mixed Graph Modelsmentioning

confidence: 99%

“…A thresholding estimator for the structure is a practical alternative to choosing the most probable graph: a difficult task for Markov chain Monte Carlo in discrete structures. An analysis of thresholding mechanisms is provided in other contexts by [1] and [4]. However, since the estimated graph might not have occurred at any point during sampling, further parameter sampling conditioned on this graph will be necessary in order to obtain as estimator for the covariance matrix with structural zeroes matching the missing edges.…”

Section: Illustration: Learning Measurement Error Structurementioning

confidence: 99%

A MCMC Approach for Learning the Structure of Gaussian Acyclic Directed Mixed Graphs

Silva

2013

Statistical Models for Data Analysis

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Graphical models are widely used to encode conditional independence constraints and causal assumptions, the directed acyclic graph (DAG) being one of the most common families of models. However, DAGs are not closed under marginalization: that is, if a distribution is Markov with respect to a DAG, several of its marginals might not be representable with another DAG unless one discards some of the structural independencies. Acyclic directed mixed graphs (ADMGs) generalize DAGs so that closure under marginalization is possible. In a previous work, we showed how to perform Bayesian inference to infer the posterior distribution of the parameters of a given Gaussian ADMG model, where the graph is fixed. In this paper, we extend this procedure to allow for priors over graph structures.Key words: Graphical models, Bayesian inference, sparsity. Acyclic Directed Mixed Graph ModelsDirected acyclic graphs (DAGs) provide a practical language to encode conditional independence constraints (see, e.g., [8]). However, such a family is not closed under marginalization. As an illustration of this concept, consider the following DAG:This model entails several conditional independencies. For instance, it encodes constraints such as Y 2 ⊥ ⊥ Y 4 , as well asDirected graphical models are non-monotonic independence models, in the sense that conditioning on extra variables can destroy and re-create independencies, as the sequence { / 0, {Y 3 }, {Y 3 , X}} has demonstrated.

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“…Um outro método que utiliza distribuições a priori de encolhimentoé o estimador Horseshoe, proposto por Carvalho et al (2010). Esses autores propuseram uma distribuição a priori hierárquica, denominada de priori Horseshoe para estimar coeficientes de regressão.…”

Section: Introductionunclassified

Modelo hierárquico bayesiano na determinação de associação entre marcadores e QTL em uma população F2.

et al. 2018

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O objetivo do mapeamento de QTL (Quantitative Trait Loci} é identificar sua posição no genoma, isto é, identificar em qual cromossomo está e qual sua localização nesse cromossomo, bem como estimar seus efeitos genéticos. Uma vez que as localizações dos QTL não são conhecidas a priori, marcadores são usados frequentemente para auxiliar no seu mapeamento. Alguns marcadores podem estar altamente ligados a um ou mais QTL e, dessa forma eles podem mostrar uma alta associação com a característica fenotípica. Os valores fenotípicos de uma característica quantitativa são normalmente descritos por um modelo linear. Geralmente, um grande número de marcadores são utilizados no modelo linear no processo de associação e dessa forma o modelo especificado contém um número elevado de parâmetros a serem estimados. No entanto, é esperado que muitos destes parâmetros sejam não significativos, necessitando de um tratamento especial. Neste trabalho é proposta a utilização de dois modelos que utilizam distribuições a priori de encolhimento: Lasso Bayesiano e o Estimador Horseshoe. Para avaliar o desempenho dos modelos na determinação da associação entre marcadores e QTL, realizou-se um estudo de simulação. Foi analisada a associação entre marcadores e QTL utilizando a característica fenotípica produção de grãos. A implementação computacional dos algoritmos foi feita utilizando a linguagem C e executada no pacote estatístico R.

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The horseshoe estimator for sparse signals

Cited by 1,207 publications

References 18 publications

Shrinkage priors for Bayesian penalized regression.

Shrinkage priors for Bayesian penalized regression.

A MCMC Approach for Learning the Structure of Gaussian Acyclic Directed Mixed Graphs

Modelo hierárquico bayesiano na determinação de associação entre marcadores e QTL em uma população F2.

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