Analytical Approximations to Conditional Distribution Functions

DiCiccio, Thomas J.; Martin, Michael; Young, George A.

doi:10.2307/2336869

Cited by 6 publications

(7 citation statements)

References 6 publications

(9 reference statements)

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“…For these cases, marginalization would require methods for multidimensional integration that we have been trying to avoid. One attractive alternative is the application of asymptotic methods of approxi‐mation (Tierney et al 1989; DiCiccio et al 1993; Shun & McCullagh 1995), which to the authors’ knowledge have never been applied to Bayesian geophysical inverse problems. To overcome modest non‐linearity of the data modelling operator, one can adopt standard procedures of non‐linear optimization and linearize the function around the solution point.…”

Section: Discussionmentioning

confidence: 99%

Local Bayesian inversion: theoretical developments

Moraes

Scales

2000

Geophysical Journal International

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Summary We derive a new Bayesian formulation for the discrete geophysical inverse problem that can significantly reduce the cost of the computations. The Bayesian approach focuses on obtaining a probability distribution (the posterior distribution), assimilating three kinds of information: physical theories (data modelling), observations (data measurements) and prior information on models. Once this goal is achieved, all inferences can be obtained from the posterior by computing statistics relative to individual parameters (e.g. marginal distributions), a daunting computational problem in high dimensions. Our formulation is developed from the working hypothesis that the local (subsurface) prior information on model parameters supercedes any additional information from other parts of the model. Based on this hypothesis, we propose an approximation that permits a reduction of the dimensionality involved in the calculations via marginalization of the probability distributions. The marginalization facilitates the tasks of incorporating diverse prior information and conducting inferences on individual parameters, because the final result is a collection of 1‐D posterior distributions. Parameters are considered individually, one at a time. The approximation involves throwing away, at each step, cross‐moment information of order higher than two, while preserving all marginal information about the parameter being estimated. The main advantage of the method is allowing for systematic integration of prior information while maintaining practical feasibility. This is achieved by combining (1) probability density estimation methods to derive marginal prior distributions from available local information, and (2) the use of multidimensional Gaussian distributions, which can be marginalized in closed form. Using a six‐parameter problem, we illustrate how the proposed methodology works. In the example, the marginal prior distributions are derived from the application of the principle of maximum entropy, which allows one to solve the entire problem analytically. Both random and modelling errors are considered. The uncertainty measure for estimated parameters is provided by 95 per cent probability intervals calculated from the marginal posterior distributions.

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Section: Discussionmentioning

confidence: 99%

Local Bayesian inversion: theoretical developments

Moraes

Scales

2000

Geophysical Journal International

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“…To obtain reasonable accuracy for smaller sample sizes requires the use of more accurate confidence intervals, either a second‐order‐accurate bootstrap interval, or comparable second‐order‐accurate non‐bootstrap interval. Two general second‐order‐accurate procedures that do not require sampling are ABC 24 and automatic percentile 25 intervals, which are approximations for BCa and tilting intervals, respectively.…”

Section: Bootstrap Confidence Intervalsmentioning

confidence: 99%

Bootstrap

Hesterberg

2011

WIREs Computational Stats

213

101

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This article provides an introduction to the bootstrap. The bootstrap provides statistical inferences—standard error and bias estimates, confidence intervals, and hypothesis tests—without assumptions such as Normal distributions or equal variances. As such, bootstrap methods can be remarkably more accurate than classical inferences based on Normal or t distributions. The bootstrap uses the same basic procedure regardless of the statistic being calculated, without requiring the use of application‐specific formulae. This article may provide two big surprises for many readers. The first is that the bootstrap shows that common t confidence intervals are woefully inaccurate when populations are skewed, with one‐sided coverage levels off by factors of two or more, even for very large samples. The second is that the number of bootstrap samples required is much larger than generally realized. WIREs Comp Stat 2011 3 497–526 DOI: 10.1002/wics.182 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bootstrap and Resampling

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“…In practical applications where F n ∣1,…, n −1 is unknown, the conditional distributions must be estimated from the sample. This can be done empirically by discretizing the space of parameters or by applying more complex methods [e.g., Diciccio et al , 1993; Hall et al , 1999]. In each case, the inverse transformation is approximated using a correspondence table between the original and the transformed variables, which implies that simulated fields will in the end be composed of measured values only.…”

Section: Modeling the Dsdmentioning

confidence: 99%

Geostatistical simulation of two‐dimensional fields of raindrop size distributions at the meso‐γ scale

Schleiss

Berne

Uijlenhoet

2009

Water Resources Research

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[1] The large variability of the raindrop size distribution (DSD) in space and time must be taken into account to improve remote sensing of precipitation. The ability to simulate a large number of 2-D fields of DSDs sharing the same statistical properties provides a very useful simulation framework that nicely complements experimental approaches based on DSD ground measurements. These simulations can be used to investigate radar beam propagation through rain and to evaluate different radar retrieval techniques. The proposed approach uses geostatistical methods to provide structural analysis and stochastic simulation of DSD fields. First, the DSD is assumed to follow a Gamma distribution with three parameters. As a consequence, 2-D fields of DSDs can be described as a multivariate random function. The parameters are normalized using a Gaussian anamorphosis and simulated by taking advantage of fast Gaussian simulation algorithms. Variograms are used to characterize the spatial structure of the DSD fields. The generated fields have identical spatial structure and are consistent with the observations. Because intermittency cannot be simulated using this technique, the size of the simulation domain is limited to the meso-g scale (2-20 km). To assess the proposed approach, the method is applied to data collected during intense Mediterranean rainfall. Taylor's hypothesis is invoked to convert time series into 1-D range profiles. The anisotropy of the fields is derived from radar measurements. Simulated and measured reflectivity fields are in good agreement with respect to the mean, the standard deviation, and the spatial structure, demonstrating the promising potential of the proposed stochastic model of DSD fields.

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Analytical Approximations to Conditional Distribution Functions

Cited by 6 publications

References 6 publications

Local Bayesian inversion: theoretical developments

Local Bayesian inversion: theoretical developments

Bootstrap

Geostatistical simulation of two‐dimensional fields of raindrop size distributions at the meso‐γ scale

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