Bayesian Point Of View Research Articles

Wavelet-Based Priors Accelerate Maximum-a-Posteriori Optimization in Bayesian Inverse Problems

Wavelet (Besov) priors are a promising way of reconstructing indirectly measured fields in a regularized manner. We demonstrate how wavelets can be used as a localized basis for reconstructing permeability fields with sharp interfaces from noisy pointwise pressure field measurements in the context of the elliptic inverse problem. For this we derive the adjoint method of minimizing the Besov-norm-regularized misfit functional (this corresponds to determining the maximum a posteriori point in the Bayesian point of view) in the Haar wavelet setting. As it turns out, choosing a wavelet–based prior allows for accelerated optimization compared to established trigonometrically–based priors.

New prior distribution for Bayesian neural network and learning via Hamiltonian Monte Carlo

A prior distribution of weights for Multilayer feedforward neural network in Bayesian point of view plays a central role toward generalization. In this context, we propose a new prior law for weights parameters which motivate the network regularization more than $$l_ {1}$$ and $$l_ {2}$$ early proposed. To train the network, we have based on Hamiltonian Monte Carlo, it is used to simulate the prior and the posterior distribution. The generated samples are used to approximate the gradient of the evidence which allows to re-estimate the hyperparameters that balance a trade off between the likelihood term and regularized term, on the other hand we use the obtained posterior samples to estimate the network output. The case problem studied in this paper includes a regression and classification tasks. The obtained results illustrate the advantages of our approach in term of error rate compared to old approach, unfortunately our method consomme time before convergence.

A flexible procedure for formulating probability distributions on the unit interval with applications

In this paper, we present a flexible mechanism for constructing probability distributions on a bounded intervals which is based on the composition of the baseline cumulative probability function and the quantile transformation from another cumulative probability distribution. In particular, we are interested in the (0, 1) intervals. The composite quantile family of probability distributions contains many models that have been proposed in the recent literature and new probability distributions are introduced on the unit interval. The proposed methodology is illustrated with two examples to analyze a poverty dataset in Peru from the Bayesian paradigm and Likelihood points of view.

Weighted Analogue of Inverse Gamma Distribution: Statistical Properties, Estimation and Simulation Study

In this article we propose a new weighted version of inverse Gamma distribution known as Weighted Inverse Gamma distribution (WIGD). We examine the Length biased and Area biased versions of Weighted Inverse Gamma distribution. Basic structural properties viz moments, mode, moment generating function (mgf), characteristic function (cf), hazard rate function and measures of uncertainty. The parameters of this model are estimated from both classical (namely, maximum likelihood estimator and method of moments, and compare them by using extensive numerical simulations) and Bayesian point of view. The Bayes estimates are estimated by using non-informative Jeffrey’s prior and informative Inverse Chi square prior under different types of loss function (symmetric and asymmetric loss functions). Finally, a simulation study has been conducted for comparing weighted inverse gamma distribution with other competing distributions.

Global analyses of Higgs portal singlet dark matter models using GAMBIT

We present global analyses of effective Higgs portal dark matter models in the frequentist and Bayesian statistical frameworks. Complementing earlier studies of the scalar Higgs portal, we use GAMBIT to determine the preferred mass and coupling ranges for models with vector, Majorana and Dirac fermion dark matter. We also assess the relative plausibility of all four models using Bayesian model comparison. Our analysis includes up-to-date likelihood functions for the dark matter relic density, invisible Higgs decays, and direct and indirect searches for weakly-interacting dark matter including the latest XENON1T data. We also account for important uncertainties arising from the local density and velocity distribution of dark matter, nuclear matrix elements relevant to direct detection, and Standard Model masses and couplings. In all Higgs portal models, we find parameter regions that can explain all of dark matter and give a good fit to all data. The case of vector dark matter requires the most tuning and is therefore slightly disfavoured from a Bayesian point of view. In the case of fermionic dark matter, we find a strong preference for including a CP-violating phase that allows suppression of constraints from direct detection experiments, with odds in favour of CP violation of the order of 100:1. Finally, we present DDCalc2.0.0, a tool for calculating direct detection observables and likelihoods for arbitrary non-relativistic effective operators.

Bayesian Generalized Two-way ANOVA Modeling for Functional Data Using INLA

Functional analysis of variance (ANOVA) modeling has been proved particularly useful to investigate the dynamic changes of functional data according to certain categorical factors and their interactions. The current existing methods often encounter difficulties when the functional data are high-dimensional, non-Gaussian, and/or exhibit certain shape characteristics that vary with spatial locations. In this paper, we investigate the functional two-way ANOVA models from a novel Bayesian perspective. A class of highly flexible Gaussian Markov random fields (GMRF) are taken as priors on the functions in the model, which allows us to model various types of functional effects, such as (discrete or continuous) temporal effects and (point-level or areal) spatial effects. The resulting posterior distributions are obtained by an efficient computational tool based on integrated nested Laplace approximations (INLA) Rue, Martino and Chopin (2009). We then employ the excursion method introduced by Bolin and Lindgren (2015) to build simultaneous credible intervals of functional effects and test their significance from a Bayesian point of view. A simulation study and multiple data examples are presented to demonstrate the merits of our method.

Characteristic values of geotechnical parameters in Eurocode 7

Lack of harmonisation between reliability-based design and the partial factor method in Eurocode 7 (EN 1997-1:2004) is preventing the widespread introduction of a risk-based concept in geotechnical design. This paper discusses how uncertainties are managed according to EN 1997-1:2004 and possible implications of not harmonising the current safety format with reliability-based design. One of several challenges highlighted is how EN 1997-1:2004 defines the characteristic value and design value. The characteristic value is therein defined based on a classical frequentist approach through a confidence interval. From a Bayesian point of view, the current definition does not treat the characteristic value as an uncertain variable. Consequently, the definitions of the characteristic value and design value in EN 1997-1:2004 feature weak connections between uncertainties in the geotechnical properties and the consequences of failure, as regulated by the target reliability index.

On progressively censored inverted exponentiated Rayleigh distribution

ABSTRACTIn this paper, we discuss a progressively censored inverted exponentiated Rayleigh distribution. Estimation of unknown parameters is considered under progressive censoring using maximum likelihood and Bayesian approaches. Bayes estimators of unknown parameters are derived with respect to different symmetric and asymmetric loss functions using gamma prior distributions. An importance sampling procedure is taken into consideration for deriving these estimates. Further highest posterior density intervals for unknown parameters are constructed and for comparison purposes bootstrap intervals are also obtained. Prediction of future observations is studied in one- and two-sample situations from classical and Bayesian viewpoint. We further establish optimum censoring schemes using Bayesian approach. Finally, we conduct a simulation study to compare the performance of proposed methods and analyse two real data sets for illustration purposes.

A Bayesian-weighted approach to predicting the number of newly discovered rare species.

In natural ecological communities, most species are rare and thus susceptible to extinction. Consequently, the prediction and identification of rare species are of enormous value for conservation purposes. How many newly found species will be rare in the next field survey? We took a Bayesian viewpoint and used observed species abundance information in an ecological sample to develop an accurate way to estimate the number of new rare species (e.g., singletons, doubletons, and tripletons) in an additional unknown sample. A similar method has been developed for incidence-based data sets. Five seminumerical tests (3 abundance cases and 2 incidence cases) showed that our proposed Bayesian-weight estimator accurately predicted the number of new rare species with low relative bias and low relative root mean squared error and, accordingly, high accuracy. Finally, we applied the proposed estimator to 6 conservation-directed empirical data sets (3 abundance cases and 3 incidence cases) and found the prediction of new rare species was quite accurate; the 95% CI covered the true observed value very well in most cases. Our estimator performed similarly to or better than an unweighted estimator derived from Chao etal. and performed consistently better than the naïve unweighted estimator. We recommend our Bayesian-weight estimator for conservation applications, although the unweighted estimator of Chao etal. may be better under some circumstances. We provide an R package RSE (rare species estimation) at https://github.com/ecomol/RSE for implementation of the estimators.

BAYESIAN INFERENCE FOR THE TANGENT PORTFOLIO

In this paper, we consider the estimation of the weights of tangent portfolios from the Bayesian point of view assuming normal conditional distributions of the logarithmic returns. For diffuse and conjugate priors for the mean vector and the covariance matrix, we derive stochastic representations for the posterior distributions of the weights of tangent portfolio and their linear combinations. Separately, we provide the mean and variance of the posterior distributions, which are of key importance for portfolio selection. The analytic results are evaluated within a simulation study, where the precision of coverage intervals is assessed.

Inference on Marshall–Olkin Bivariate Exponential in the Presence of Dependent Left Censoring

This paper deals with the dependent left censoring scheme when the survival time variable and censoring variable are dependent and have Marshal–Olkin bivariate exponential distribution. We use the expectation–conditional maximization algorithm for finding the maximum likelihood estimates of the unknown parameters. From Bayesian point of view, based on a particular choice of hyperparameters of prior distribution we obtain the exact Bayes estimates of the unknown parameters. We employ importance sampling MCMC technique and an approximate Bayes estimation method to compute Bayes estimates of the unknown parameters. Finally, a Monte Carlo simulation and a real data analysis are studied.

Rationality under uncertainty: classic and current criticisms of the Bayesian viewpoint

At least since Leonard Savage’s extension of von Neumann and Morgenstern’s expected utility, rational choice theory has been interpreted as a theory prescribing what individuals should do in any decision context, ranging from certainty to risk and uncertainty. After decades this received view, usually termed Bayesian, has been criticized for its normative content. This paper compares the current critique of the notion of Bayesian rationality, proposed by Itzhak Gilboa, with Daniel Ellsberg’s classic critique of Savage’s understanding of rationality. The paper argues that Ellsberg’s classic analysis of Savage’s theory totally anticipated today’s criticism.

A regularization method for constructing trend function in Kriging model

Kriging is a popular surrogate for approximating computationally expensive computer experiments. When sample points are limited, it is difficult to identify the overall trend of the problem at hand properly. Thanks to the interpolating characteristic of the Kriging model, a constant is widely used as the trend function, which neglects the overall trend presented by data. However, previous researches prove that an appropriate trend function considering high-order terms is able to enhance the approximation ability of the Kriging model. In this paper, a regularization approach is proposed to construct the trend function in the Kriging model to improve the prediction accuracy. First, a new weighting scheme, which is formulated as an optimization problem with regularization terms, is used to solve the regression coefficients. Then, the other model parameters are estimated by maximizing the likelihood function, which leads to a nested optimization problem. It is solved iteratively to obtain the final estimation of the model parameters. From a Bayesian point of view, the proposed regularization method can adaptively tune the parameter of the prior distribution on the regression coefficients in the iterative algorithm. To select good regularization parameters, a cross-validation method is used. The implementation is tested on several analytical examples and physical examples, and the experimental results confirm the effectiveness of the proposed approach.

Objective and subjective prior distributions for the Gompertz distribution.

This paper takes into account the estimation for the unknown parameters of the Gompertz distribution from the frequentist and Bayesian view points by using both objective and subjective prior distributions. We first derive non-informative priors using formal rules, such as Jefreys prior and maximal data information prior (MDIP), based on Fisher information and entropy, respectively. We also propose a prior distribution that incorporate the expert's knowledge about the issue under study. In this regard, we assume two independent gamma distributions for the parameters of the Gompertz distribution and it is employed for an elicitation process based on the predictive prior distribution by using Laplace approximation for integrals. We suppose that an expert can summarize his/her knowledge about the reliability of an item through statements of percentiles. We also present a set of priors proposed by Singpurwala assuming a truncated normal prior distribution for the median of distribution and a gamma prior for the scale parameter. Next, we investigate the effects of these priors in the posterior estimates of the parameters of the Gompertz distribution. The Bayes estimates are computed using Markov Chain Monte Carlo (MCMC) algorithm. An extensive numerical simulation is carried out to evaluate the performance of the maximum likelihood estimates and Bayes estimates based on bias, mean-squared error and coverage probabilities. Finally, a real data set have been analyzed for illustrative purposes.

Fully Bayesian Prediction Algorithms for Mobile Robotic Sensors under Uncertain Localization Using Gaussian Markov Random Fields.

In this paper, we present algorithms for predicting a spatio-temporal random field measured by mobile robotic sensors under uncertainties in localization and measurements. The spatio-temporal field of interest is modeled by a sum of a time-varying mean function and a Gaussian Markov random field (GMRF) with unknown hyperparameters. We first derive the exact Bayesian solution to the problem of computing the predictive inference of the random field, taking into account observations, uncertain hyperparameters, measurement noise, and uncertain localization in a fully Bayesian point of view. We show that the exact solution for uncertain localization is not scalable as the number of observations increases. To cope with this exponentially increasing complexity and to be usable for mobile sensor networks with limited resources, we propose a scalable approximation with a controllable trade-off between approximation error and complexity to the exact solution. The effectiveness of the proposed algorithms is demonstrated by simulation and experimental results.

Bayesian Estimation of Beta-type Distribution Parameters Based on Grouped Data

This study considers a method of estimating generalized beta (GB) distribution parameters based on grouped data from a Bayesian point of view and explores the possibility of the GB distribution focusing on the goodness-of-fit because the GB distribution is one of the most typical five-parameter distributions. It uses a tailored randomized block Metropolis–Hastings (TaRBMH) algorithm to estimate the GB distribution parameters and this method is then applied to one simulated and two real datasets. Moreover, the fit of the GB distribution is compared with those of the generalized beta distribution of the second kind (GB2 distribution) and Dagum (DA) distribution by using the marginal likelihood. The estimation results of simulated and real datasets show that the GB distributions are preferred to the DA distributions in general, while the GB2 distributions have similar performances to the GB distributions. In other words, the GB2 distribution could be adopted as well as the GB distribution in terms of the smallest possible number of parameters, although our TaRBMH algorithm can estimate the GB distribution parameters efficiently and accurately. The accuracy of the Gini coefficients also suggests the use of the GB2 distributions.

Statistical properties and different methods of estimation of Gompertz distribution with application

This article addresses the various properties and different methods of estimation of the unknown parameters of Gompertz distribution. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of the Gompertz distribution (such as quantiles, moments, moment generating function, hazard rate, mean residual lifetime, mean past lifetime, stochasic ordering, stress-strength parameter, various entropies, Bonferroni and Lorenz curves and order statistics) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, moments estimators, pseudo-moments estimators, modified moments estimators, L-moment estimators, percentile based estimators, least squares and weighted least squares estimators, maximum product of spacings estimators, minimum spacing absolute distance estimators, minimum spacing absolute-log distance estimator, Cramér-von-Mises estimators, Anderson-Darling and right-tail Anderson-Darling and compare them using extensive numerical simulations. Coverage probabilities for the frequentist methods are also obtained. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using gamma priors for both shape and scale parameters. Furthermore, the Bayes estimators and their respective posterior risks are computed and compared using MCMC algorithm. Finally, a real data set have been analyzed for illustrative purposes.

Comment: The Inferential Information Criterion from a Bayesian Point of View

Comment: The Inferential Information Criterion from a Bayesian Point of View

Converting nonrelativistic dark matter to radiation

Dark matter in the cosmological concordance model is parameterised by a single number, describing the covariantly conserved energy density of a non-relativistic fluid. Here we test this assumption in a model-independent and conservative way by considering the possibility that, at any point during the cosmological evolution, dark matter may be converted into a non-interacting form of radiation. This scenario encompasses, but is more general than, the cases where dark matter decays or annihilates into these states. We show that observations of the cosmic microwave background allow to strongly constrain this scenario for any conversion time after big bang nucleosynthesis. We discuss in detail, both from a Bayesian and frequentist point of view, in which sense adding large-scale structure observations may even provide a certain preference for a conversion of dark matter to radiation at late times. Finally we apply our general results to a specific particle physics realisation of such a scenario, featuring late kinetic decoupling and Sommerfeld-enhanced dark matter annihilation. We identify a small part of parameter space that both mitigates the tension between cosmic microwave and large-scale structure data and allows for velocity-dependent dark matter self-interactions strong enough to address the small-scale problems of structure formation.

Application of a non-homogeneous Markov chain with seasonal transition probabilities to ozone data

In this work, we assume that the sequence recording whether or not an ozone exceedance of an environmental threshold has occurred in a given day is ruled by a non-homogeneous Markov chain of order one. In order to account for the possible presence of cycles in the empirical transition probabilities, a parametric form incorporating seasonal components is considered. Results show that even though some covariates (namely, relative humidity and temperature) are not included explicitly in the model, their influence is captured in the behavior of the transition probabilities. Parameters are estimated using the Bayesian point of view via Markov chain Monte Carlo algorithms. The model is applied to ozone data obtained from the monitoring network of Mexico City, Mexico. An analysis of how the methodology could be used as an aid in the decision-making is also given.