Exact Bayesian inference for the Bingham distribution

Fallaize, Christopher J.; Kypraios, Theodore

doi:10.1007/s11222-014-9508-7

Cited by 15 publications

(23 citation statements)

References 25 publications

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“…As pointed out by Kent et al (2013), evaluating the acceptance probability is difficult because it depends on the normalizing constant; however, it has been verified empirically that the efficiency is never lower than 52% when q = 3 (Kent et al, 2013). For larger q, the efficiency deteriorates rather quickly; although the actual acceptance rate depends on the numerical values of the parameters, when q = 7 some simulations whose results are not reported here give an average acceptance probability close to the 10% found by Fallaize and Kypraios (2016). Hence, AMLE becomes computationally more demanding for large-dimensional problems; see Section 4.3 for further details.…”

Section: Amle Of the Bingham Distributionmentioning

confidence: 73%

“…The Bingham distribution can be simulated by means of an accept-reject algorithm (Kent et al, 2013; see also Fallaize and Kypraios, 2016) that uses the Angular Central Gaussian distribution (ACG; Tyler, 1987) as an envelope. As pointed out by Kent et al (2013), evaluating the acceptance probability is difficult because it depends on the normalizing constant; however, it has been verified empirically that the efficiency is never lower than 52% when q = 3 (Kent et al, 2013).…”

Section: Amle Of the Bingham Distributionmentioning

confidence: 99%

“…Recently, the properties of the large dimensional Bingham distribution have been studied by Kume and Walker (2014). The need of modeling such data arises in many scientific fields, such as geology (Peel et al, 2001), crystallography (Krieger Lassen et al, 1994) and bioinformatics (Kent and Hamelryck, 2005;Hamelryck et al, 2006;Boomsma et al, 2008); see also Mardia and Jupp (2000) or Fallaize and Kypraios (2016) and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Approximate maximum likelihood estimation of the Bingham distribution

Bee

Benedetti

Espa

2017

Computational Statistics & Data Analysis

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Maximum likelihood estimation of the Bingham distribution is difficult because the density function contains a normalization constant that cannot be computed in closed form. Given the availability of sufficient statistics, Approximate Maximum Likelihood Estimation (AMLE) is an appealing method that allows one to bypass the evaluation of the likelihood function. The impact of the input parameters of the AMLE algorithm is investigated and some methods for choosing their numerical values are suggested. Moreover, AMLE is compared to the standard approach which numerically maximizes the (approximate) likelihood obtained with the normalization constant estimated via the Holonomic Gradient Method (HGM). For the Bingham distribution on the sphere, simulation experiments and real-data applications produce similar outcomes for both methods. On the other hand, AMLE outperforms HGM when the dimension increases.

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Section: Amle Of the Bingham Distributionmentioning

confidence: 73%

Section: Amle Of the Bingham Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate maximum likelihood estimation of the Bingham distribution

Bee

Benedetti

Espa

2017

Computational Statistics & Data Analysis

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“…So we have three classes of directional data where Bingham distributions are considered appropriate. In particular, a Bayesian modelling approach to fitting Bingham distributions to such data is also considered in Fallaize and Kypraios (2014). We will show below that in fact the best modelling choice among the sub-classes of Fisher-Bingham family is indeed the Bingham distribution.…”

Section: Numerical Evidencementioning

confidence: 99%

On the exact maximum likelihood inference of Fisher–Bingham distributions using an adjusted holonomic gradient method

Kume

Sei

2017

Stat Comput

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Holonomic function theory has been successfully implemented in a series of recent papers to efficiently calculate the normalizing constant and perform likelihood estimation for the Fisher-Bingham distributions. A key ingredient for establishing the standard holonomic gradient algorithms is the calculation of the Pfaffian equations. So far, these papers either calculate these symbolically or apply certain methods to simplify this process. Here we show the explicit form of the Pfaffian equations using the expressions from Laplace inversion methods. This improves on the implementation of the holonomic algorithms for these problems and enables their adjustments for the degenerate cases. As a result, an exact and more dimensionally efficient ODE is implemented for likelihood inference.

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“…Simulation algorithms for distributions of the Fisher-Bingham family have been implemented byPapadakis et al (2018) in an open-source R package Rfast. Open-source tools for statistical simulation in the R and Python environments (including their combined usage through the rypy2 package), provide convenient, well-documented tools for applying established statistical techniques to novel fields in geoscience.The algorithm for simulating random points from the Bingham and Kent distributions included in Rfast uses the acceptancerejection method, inspired byKent et al (2013) andFallaize and Kypraios (2016). The method uses a Central Angular Gaussian (CAG) distribution as an envelope to approximate the Bingham distribution.…”

mentioning

confidence: 99%

Uncertainty assessment for 3D geologic modeling of fault zones based on geologic inputs and prior knowledge

Krajnovich¹,

Zhou²,

Gutierrez³

2020

Preprint

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Abstract. Characterizing the zone of damaged and altered rock surrounding a fault surface is highly relevant to geotechnical and reservoir engineering works in the subsurface. Evaluating the uncertainty associated with 3D geologic modeling of these fault zones is made possible using the popular and flexible input-based uncertainty propagation approach to geologic model uncertainty assessment – termed Monte Carlo simulation for uncertainty propagation (MCUP). To satisfy the automation requirements of MCUP while still preserving the key geometry of fault zones in the subsurface, a clear and straightforward modeling approach is developed based on four geologic inputs used in implicit geologic modeling algorithms (surface trace, structural orientation, vertical termination depth and fault zone thickness). The rationale applied to identifying and characterizing the various sources of uncertainty affecting each input are explored and provided using open-source codes. In considering these sources of uncertainty, a novel model formulation is implemented using prior geologic knowledge (i.e., empirical and theoretical relationships) to parameterize modeling inputs which are typically subjectively interpreted by the modeler (e.g., vertical termination depth of fault zones). Additionally, the application of anisotropic spherical distributions to modeling disparate levels of information available regarding a fault zone's dip azimuth and dip angle is demonstrated, providing improved control over the structural orientation uncertainty envelope. The MCUP formulation developed is applied to a simple geologic model built from historically available geologic mapping data to assess the independent sensitivity of each modeling input on the combined model uncertainty, revealing that vertical termination depth and structural orientation uncertainty dominate model uncertainty at depth while surface trace uncertainty dominates model uncertainty near the ground surface. The method is also successfully applied to a more complex model containing intersecting major and minor fault zones. The impacts of the model parameterization choices, the fault zone modeling approach and the effects of fault zone interactions on the final geologic model uncertainty assessment are discussed.

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Exact Bayesian inference for the Bingham distribution

Cited by 15 publications

References 25 publications

Approximate maximum likelihood estimation of the Bingham distribution

Approximate maximum likelihood estimation of the Bingham distribution

On the exact maximum likelihood inference of Fisher–Bingham distributions using an adjusted holonomic gradient method

Uncertainty assessment for 3D geologic modeling of fault zones based on geologic inputs and prior knowledge

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