Saddlepoint approximations for the normalizing constant of Fisher-Bingham distributions on products of spheres and Stiefel manifolds

Kume, Alfred; Preston, Simon; Wood, Andrew

doi:10.1093/biomet/ast021

Cited by 37 publications

(34 citation statements)

References 16 publications

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“…Table 1 shows the results of a simulation study, including computational timings, for fitting ESAG and Kent densities to ESAG and Kent simulated data. To approximate the Kent normalising constant when fitting the Kent distribution, we use a saddlepoint approximation method (see Kume et al 2013), and for simulating from the Kent distribution, we use the rejection method of Kent et al (2013). A notion of accuracy of the fitted model is how well the mean direction of the fitted model,m, corresponds with the population mean direction, m. A measure we use for this is…”

Section: An Example: Estimates Of the Historic Position Of Earth's Mamentioning

confidence: 99%

An elliptically symmetric angular Gaussian distribution

et al. 2017

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We define a distribution on the unit sphere S d−1 called the elliptically symmetric angular Gaussian distribution. This distribution, which to our knowledge has not been studied before, is a subfamily of the angular Gaussian distribution closely analogous to the Kent subfamily of the general Fisher-Bingham distribution. Like the Kent distribution, it has ellipse-like contours, enabling modelling of rotational asymmetry about the mean direction, but it has the additional advantages of being simple and fast to simulate from, and having a density and hence likelihood that is easy and very quick to compute exactly. These advantages are especially beneficial for computationally intensive statistical methods, one example of which is a parametric bootstrap procedure for inference for the directional mean that we describe.

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Section: An Example: Estimates Of the Historic Position Of Earth's Mamentioning

confidence: 99%

An elliptically symmetric angular Gaussian distribution

et al. 2017

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“…f) Matrix Bingham-von Mises-Fisher distribution (BMF): To cope with orientation, for example represented as rotation matrix, Matrix Bingham-von Mises-Fisher distribution (BMF) [28] can be used. Its normalizing constant is intractable and requires approximation [29], which is not a problem in our case, as we integrate over robot configurations. Its density…”

Section: Samples Ofmentioning

confidence: 99%

Variational Inference with Mixture Model Approximation for Applications in Robotics

Pignat¹,

Lembono²,

Calinon³

2020

2020 IEEE International Conference on Robotics and Automation (ICRA)

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We propose to formulate the problem of representing a distribution of robot configurations (e.g. joint angles) as that of approximating a product of experts. Our approach uses variational inference, a popular method in Bayesian computation, which has several practical advantages over sampling-based techniques. To be able to represent complex and multimodal distributions of configurations, mixture models are used as approximate distribution. We show that the problem of approximating a distribution of robot configurations while satisfying multiple objectives arises in a wide range of problems in robotics, for which the properties of the proposed approach have relevant consequences. Several applications are discussed, including learning objectives from demonstration, planning, and warm-starting inverse kinematics problems. Simulated experiments are presented with a 7-DoF Panda arm and a 28-DoF Talos humanoid.

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“…In comparison, the ESAG density, (8), is rather cumbersome. On the other hand, the ESAG density and likelihood can be computed exactly, whereas the Kent density and likelihood involves a normalising constant, C(κ, β) in 5, which is not known in closed form and hence needs to be approximated, by truncating an infinite series (Kent 1982), or else by saddlepoint or holonomic gradient methods (Kume and Sei 2017;Kume et al 2013). In the present context, we maximise the likelihood for the regression models numerically, so the ESAG likelihood having a cumbersome form is no drawback, and the fact that it can be computed exactly is an advantage.…”

Section: Practical Differences Between Kent and Esag Distributionsmentioning

confidence: 99%

Spherical regression models with general covariates and anisotropic errors

et al. 2019

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Existing parametric regression models in the literature for response data on the unit sphere assume that the covariates have particularly simple structure, for example that they are either scalar or are themselves on the unit sphere, and/or that the error distribution is isotropic. In many practical situations, such models are too inflexible. Here, we develop richer parametric spherical regression models in which the covariates can have quite general structure (for example, they may be on the unit sphere, in Euclidean space, categorical or some combination of these) and in which the errors are anisotropic. We consider two anisotropic error distributions-the Kent distribution and the elliptically symmetric angular Gaussian distribution-and two parametrisations of each which enable distinct ways to model how the response depends on the covariates. Various hypotheses of interest, such as the significance of particular covariates, or anisotropy of the errors, are easy to test, for example by classical likelihood ratio tests. We also introduce new model-based residuals for evaluating the fitted models. In the examples we consider, the hypothesis tests indicate strong evidence to favour the novel models over simpler existing ones.

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Saddlepoint approximations for the normalizing constant of Fisher-Bingham distributions on products of spheres and Stiefel manifolds

Cited by 37 publications

References 16 publications

An elliptically symmetric angular Gaussian distribution

An elliptically symmetric angular Gaussian distribution

Variational Inference with Mixture Model Approximation for Applications in Robotics

Spherical regression models with general covariates and anisotropic errors

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