Fundamentals of Nonparametric Bayesian Inference

Ghosal, Subhashis; Vaart, Aad van der

doi:10.1017/9781139029834

Cited by 501 publications

(537 citation statements)

References 256 publications

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“…the space of functions on [0,1] p with bounded partial derivatives up to order β , where β is the largest integer strictly less than α and such that the partial derivatives of order β are Hölder continuous of order α − β . Let

C^{α, R} false({false[0, 1 false]}^{p} false) = false{f \in C^{α} false({false[0, 1 false]}^{p} false) {: ‖ f ‖}_{α} ⩽ R false}

denote the Hölder ball of radius R with respect to the Hölder norm ║ f ║ α (see Ghosal and van der Vaart (), appendix C).…”

Section: Theoretical Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Bayesian Regression Tree Ensembles that Adapt to Smoothness and Sparsity

Linero

Yang

2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

100

143

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Summary Ensembles of decision trees are a useful tool for obtaining flexible estimates of regression functions. Examples of these methods include gradient‐boosted decision trees, random forests and Bayesian classification and regression trees. Two potential shortcomings of tree ensembles are their lack of smoothness and their vulnerability to the curse of dimensionality. We show that these issues can be overcome by instead considering sparsity inducing soft decision trees in which the decisions are treated as probabilistic. We implement this in the context of the Bayesian additive regression trees framework and illustrate its promising performance through testing on benchmark data sets. We provide strong theoretical support for our methodology by showing that the posterior distribution concentrates at the minimax rate (up to a logarithmic factor) for sparse functions and functions with additive structures in the high dimensional regime where the dimensionality of the covariate space is allowed to grow nearly exponentially in the sample size. Our method also adapts to the unknown smoothness and sparsity levels, and can be implemented by making minimal modifications to existing Bayesian additive regression tree algorithms.

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C^{α, R} false({false[0, 1 false]}^{p} false) = false{f \in C^{α} false({false[0, 1 false]}^{p} false) {: ‖ f ‖}_{α} ⩽ R false}

denote the Hölder ball of radius R with respect to the Hölder norm ║ f ║ α (see Ghosal and van der Vaart (), appendix C).…”

Section: Theoretical Resultsmentioning

confidence: 99%

“…[0, 1] p / = {f ∈ C α . [0, 1] p / : f α R} denote the Hölder ball of radius R with respect to the Hölder norm f α (see Ghosal and van der Vaart (2017), appendix C).…”

Section: Theoretical Resultsmentioning

confidence: 99%

Bayesian Regression Tree Ensembles that Adapt to Smoothness and Sparsity

Linero

Yang

2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

100

143

View full text Add to dashboard Cite

show abstract

“…Escobar and West (1995) and MacEachern and Müller (1998) develop practical models based on the DP prior. For a more comprehensive review of the Bayesian nonparametric literature, see Ghosal and Van der Vaart (2017). mode of G and the estimated G by BIC are always close to the true value with a high probability in the repeated sampling context.…”

Section: Connections To the Literaturementioning

confidence: 97%

Hidden group patterns in democracy developments: Bayesian inference for grouped heterogeneity

Kim

Wang

2019

J of Applied Econometrics

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We propose a nonparametric Bayesian approach to estimate time-varying grouped patterns of heterogeneity in linear panel data models. Unlike the classical approach in Bonhomme and Manresa (Econometrica, 2015(Econometrica, , 83, 1147(Econometrica, -1184, our approach can accommodate selection of the optimal number of groups and model estimation jointly, and also be readily extended to quantify uncertainties in the estimated group structure. Our proposed approach performs well in Monte Carlo simulations. Using our approach, we successfully replicate the estimated relationship between income and democracy in Bonhomme and Manresa and the group characteristics when we use the same number of groups. Furthermore, we find that the optimal number of groups could depend on model specifications on heteroskedasticity and discuss ways to choose models in practice. KEYWORDSbayesian inference, dirichlet process prior, panel data JEL CLASSIFICATIONF50, C11, C14

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“…Gelman et al (2014, Part V) includes a discussion of nonparametric Bayesian data analysis. A mathematically rigorous discussion, with an emphasis on asymptotic properties can be found in the forthcoming book by Ghoshal and van der Vaart (2015).…”

Section: Example 2 (Oral Cancermentioning

confidence: 99%

“…We refer interested readers to Phadia (2013), who discusses all the same models that also feature in this text. See also Ghosh and Ramamoorthi (2003), Ghoshal (2010), and Ghoshal and van der Vaart (2015) for a mathematically more rigorous discussion. Briefly summarized, assume an underlying probability space .…”

Section: Example 2 (Oral Cancermentioning

confidence: 99%