Moment conditions and Bayesian nonparametrics

Bornn, Luke; Shephard, Neil; Solgi, Reza

doi:10.48550/arxiv.1507.08645

Cited by 5 publications

(7 citation statements)

References 51 publications

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“…Sampling from manifolds has applications to areas such as statistics [6,10], computer graphics [30], optimization [11] and systems biology [34]. For a simple example, manifolds with nonnegative curvature appear in statistical applications as the level sets of log-concave, or more generally quasiconcave, distributions (see Remark 1.1).…”

Section: Introductionmentioning

confidence: 99%

Rapid mixing of geodesic walks on manifolds with positive curvature

Mangoubi¹,

Smith²

2018

Ann. Appl. Probab.

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We introduce a Markov chain for sampling from the uniform distribution on a Riemannian manifold M, which we call the geodesic walk. We prove that the mixing time of this walk on any manifold with positive sectional curvature C x (u, v) bounded both aboveIn particular, this bound on the mixing time does not depend explicitly on the dimension of the manifold. In the special case that M is the boundary of a convex body, we give an explicit and computationally tractable algorithm for approximating the exact geodesic walk. As a consequence, we obtain an algorithm for sampling uniformly from the surface of a convex body that has running time bounded solely in terms of the curvature of the body. ‡ 2 is often very pessimistic and can be greatly improved in some important special cases (See Remark 8.11 in Section 8.2). Furthermore, in situations where one has access to better geodesic approximations, such as when symplectic integrators or exact geodesic integrators are available, our results imply computational costs that are much closer to the mixing time O * M 2 m 2 of the true geodesic walk.Remark 1.2. Note that the bound in [11] is better for chains with a bound on the diameter of K that is much smaller than 2 √ m 2 , while our bound is better in higher dimensions and when the bound on the inner radius of curvature of K is much smaller than 1 √ M 2 . Also, a practical implementation of the geodesic walk requires additional computational costs that depend on the method used to approximate geodesics.

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Section: Introductionmentioning

confidence: 99%

Rapid mixing of geodesic walks on manifolds with positive curvature

Mangoubi¹,

Smith²

2018

Ann. Appl. Probab.

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“…1 Figure 1 demonstrates that Bayesian nonparametric quantile estimation is not a special case of Bayesian nonparametric ψ type M-estimators, and so not a special case of moment estimation. This means we are outside the framework developed by Bornn et al (2016).…”

Section: Definition Of the Problemmentioning

confidence: 99%

“…For the quantile problem, with probability one β = t(θ), so the "area formula" of Federer (1969) (see also Diaconis et al (2013) and Bornn et al (2016)) implies the marginal density for the probabilities is induced as…”

Section: The Prior and Posteriormentioning

confidence: 99%

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Nonparametric hierarchical Bayesian quantiles

Bornn¹,

Shephard²,

Solgi³

2016

Preprint

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Here we develop a method for performing nonparametric Bayesian inference on quantiles. Relying on geometric measure theory and employing a Hausdorff base measure, we are able to specify meaningful priors for the quantile while treating the distribution of the data otherwise nonparametrically. We further extend the method to a hierarchical model for quantiles of subpopulations, linking subgroups together solely through their quantiles. Our approach is computationally straightforward, allowing for censored and noisy data. We demonstrate the proposed methodology on simulated data and an applied problem from sports statistics, where it is observed to stabilize and improve inference and prediction.

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“…See Yin (2009), for example, who proposes the use of the asymptotic distribution of the sample moments as an approximation of the likelihood function -an idea that is closely related to the use of a Laplace approximation as proposed by Chernozhukov and Hong (2003). In related work, and following on from earlier work by Chamberlain and Imbens (2003), Bayesian nonparametric methods are explored and further developed for moment condition models by Bornn et al (2015).…”

Section: Introductionmentioning

confidence: 99%

A Robust Bayesian Exponentially Tilted Empirical Likelihood Method

Liu,

Forbes,

Anderson

2018

Preprint

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This paper proposes a new Bayesian approach for analysing moment condition models in the situation where the data may be contaminated by outliers. The approach builds upon the foundations developed by Schennach (2005) who proposed the Bayesian exponentially tilted empirical likelihood (BETEL) method, justified by the fact that an empirical likelihood (EL) can be interpreted as the nonparametric limit of a Bayesian procedure when the implied probabilities are obtained from maximizing entropy subject to some given moment constraints. Considering the impact that outliers are thought to have on the estimation of population moments, we develop a new robust BETEL (RBETEL) inferential methodology to deal with this potential problem. We show how the BETEL methods are linked to the recent work of Bissiri et al. (2016) who propose a general framework to update prior belief via a loss function. A controlled simulation experiment is conducted to investigate the performance of the RBETEL method. We find that the proposed methodology produces reliable posterior inference for the fundamental relationships that are embedded in the majority of the data, even when outliers are present.The method is also illustrated in an empirical study relating brain weight to body weight using a dataset containing sixty-five different land animal species.

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Moment conditions and Bayesian nonparametrics

Cited by 5 publications

References 51 publications

Rapid mixing of geodesic walks on manifolds with positive curvature

Rapid mixing of geodesic walks on manifolds with positive curvature

Nonparametric hierarchical Bayesian quantiles

A Robust Bayesian Exponentially Tilted Empirical Likelihood Method

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