Bayesian Reference Analysis for the Generalized Normal Linear Regression Model

This article proposes the use of the Bayesian reference analysis to estimate the parameters of the generalized normal linear regression model. It is shown that the reference prior led to a proper posterior distribution, while the Jeffreys prior returned an improper one. The inferential purposes were obtained via Markov Chain Monte Carlo (MCMC). Furthermore, diagnostic techniques based on the Kullback–Leibler divergence were used. The proposed method was illustrated using artificial data and real data on the height and diameter of Eucalyptus clones from Brazil.

The spectral index of polarized diffuse Galactic emission between 30 and 44 GHz

ABSTRACT We present an estimate of the polarized spectral index between the Planck 30 and 44 GHz surveys in 3.7° pixels. We use an objective reference prior that maximizes the impact of the data on the posterior and multiply this by a maximum entropy prior that includes information from observations in total intensity by assuming a polarization fraction. Our parametrization of the problem allows the reference prior to be easily determined and also provides a natural method of including prior information. The spectral index map is consistent with those found by others between surveys at similar frequencies. Across the entire sky we find an average spectral index of −2.99 ± 0.03(±1.12), where the first error term is the statistical uncertainty on the mean and the second error term (in parentheses) is the extra intrinsic scatter of the spectral index across the sky. We use a clustering algorithm to identify pixels with actual detections of the spectral index. The average spectral index in these pixels is −3.12 ± 0.03(±0.64) and then when also excluding pixels within 10° of the Galactic plane we find −2.92(±0.03). We find a statistically significant difference between the average spectral indices in the north and south Fermi bubbles. Only including pixels identified by the clustering algorithm, the average spectral index in the southern bubble is −3.00 ± 0.05(±0.35), which is similar to the average across the whole sky. In the northern bubble, we find a much harder average spectral index of −2.36 ± 0.09(±0.63). Therefore, if the bubbles are features in microwave polarization they are not symmetric about the Galactic plane.

Posterior Propriety of an Objective Prior in a 4-Level Normal Hierarchical Model

The use of hierarchical Bayesian models in statistical practice is extensive, yet it is dangerous to implement the Gibbs sampler without checking that the posterior is proper. Formal approaches to objective Bayesian analysis, such as the Jeffreys-rule approach or reference prior approach, are only implementable in simple hierarchical settings. In this paper, we consider a 4-level multivariate normal hierarchical model. We demonstrate the posterior using our recommended prior which is proper in the 4-level normal hierarchical models. A primary advantage of the recommended prior over other proposed objective priors is that it can be used at any level of a hierarchical model.

A probabilistic geologic model of the Krafla geothermal system constrained by gravimetric data

The quantitative connections between subsurface geologic structure and measured geophysical data allow 3D geologic models to be tested against measurements and geophysical anomalies to be interpreted in terms of geologic structure. Using a Bayesian framework, geophysical inversions are constrained by prior information in the form of a reference geologic model and probability density functions (pdfs) describing petrophysical properties of the different lithologic units. However, it is challenging to select the probabilistic weights and the structure of the prior model in such a way that the inversion process retains relevant geologic insights from the prior while also exploring the full range of plausible subsurface models. In this study, we investigate how the uncertainty of the prior (expressed using probabilistic constraints on commonality and shape) controls the inferred lithologic and mass density structure obtained by probabilistic inversion of gravimetric data measured at the Krafla geothermal system. We combine a reference prior geologic model with statistics for rock properties (grain density and porosity) in a Bayesian inference framework implemented in the GeoModeller software package. Posterior probability distributions for the inferred lithologic structure, mass density distribution, and uncertainty quantification metrics depend on the assumed geologic constraints and measurement error. As the uncertainty of the reference prior geologic model increases, the posterior lithologic structure deviates from the reference prior model in areas where it may be most likely to be inconsistent with the observed gravity data and may need to be revised. In Krafla, the strength of the gravity field reflects variations in the thickness of hyaloclastite and the depth to high-density basement intrusions. Moreover, the posterior results suggest that a WNW–ESE-oriented gravity low that transects the caldera may be associated with a zone of low hyaloclastite density. This study underscores the importance of reliable prior constraints on lithologic structure and rock properties during Bayesian geophysical inversion.

Reference analysis of non-regular models and nonparametric Bayes modeling of large data

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] Bayesian analysis is a principled approach, which makes inference about the parameter, by combining the information gained from the data and the prior belief about the parameter. There's no convergence on the choice of priors, and often different motivations for prior lead to different areas of study in Bayesian statistics. This work is motivated by two such choices, namely: reference priors and nonparametric priors. Reference priors arise out of the need to specify priors in a non-subjective manner, i.e. objective manner. Reference priors maximize the amount of information gained from the data about the parameter, in information theoretical sense. The appeal of reference priors lies in the fact that it has nice frequentist properties even for small sample size and often avoids marginalization paradoxes in Bayesian analysis. However, reference prior algorithms are typically available when the posterior is asymptotically normal and Fisher's information matrix is well-defined. In statistical parlance, such models are called regular case or regular model. Recently, Berger et al. (2009) [1] proposed a general expression of reference prior for single continuous parameter model, which is applicable for both regular and non-regular case. Motivated by Berger et al. (2009) [1], we explore reference prior methodology for a general model. Specifically, we derive expression of reference prior for single continuous parameter truncated exponential family and a general expression of conditional reference prior for multi group continuous parameter model. Furthermore, we demonstrate the usefulness of our work by deriving reference priors for models which have no known existing reference priors. We also extend Datta and Ghosh (1996) [2]'s invariance result for reference prior of regular model to general model. Nonparametric priors arise out of the need to specify priors over a large support.

Objective Bayesian analysis of the 2 x 2 contingency table and the negative binomial distribution

In Bayesian analysis, the “objective” Bayesian approach seeks to select a prior distribution not by using (often subjective) scientific belief or by mathematical convenience, but rather by deriving it under a pre-specified criteria. This approach takes the decision of prior selection out of the hands of the researcher. Ideally, for a given data model, we would like to have a prior which represents a "neutral" prior belief in the phenomenon we are studying. In categorical data analysis, the odds ratio is one of several approaches to quantify how strongly the presence or absence of one property is associated with the presence or absence of another property. In this project, we present a Reference prior for the odds ratio of an unrestricted 2 x 2 table. Posterior simulation can be conducted without MCMC and is implemented on a GPU via the CUDA extensions for C. Simulation results indicate that the proposed approach to this problem is far superior to the widely used Frequentist approaches that dominate this area. Real data examples also typically yield much more sensible results, especially for small sample sizes or for tables that contain zeros. An R package is also presented to allow for easy implementation of this methodology. Next, we develop an approximate reference prior for the negative binomial distribution, applying this methodology to a continuous parameterization often used for modeling over-dispersed count data as well as the typical discrete case. Results indicate that the developed prior equals the performance of the MLE in estimating the mean of the distribution but is far superior when estimating the dispersion parameter.

Contribution of Informed Reference Prior Using Primary and Summary Measure Data in Bayesian Meta-Analysis

When meta-analysis includes a small number of trials, inferences are sensitive to the choice of prior distributions for between-study heterogeneity. The common practice is to use vague prior but inferences depend on the degree of vagueness. Pullenayegum (2011) proposed an informed reference prior for between-study heterogeneity of binary outcomes. We employ a model for applying this prior for both primary outcome and summary measure data in Bayesian meta-analysis. We have found the same inference using both primary outcome and summary measure data. This study also suggests that the informed reference prior for between-study heterogeneity represents more relevant conclusion as compared to commonly used prior distributions.Dhaka Univ. J. Sci. 63(2): 61-65, 2015 (July)