Bayesian Bandwidth Estimation in Nonparametric Time-Varying Coefficient Models

Bayesian Bandwidth Selection in Nonparametric Time-Varying Coefficient Models

There are various analysis methods for CT perfusion (CTP). Although the advantages of Bayesian estimation algorithms have been newly suggested, comparisons with other analysis methods on clinical data are still limited. In this study, we compared the Bayesian estimation method with the singular value decomposition (SVD) method in the evaluation of patients with acute cerebral infarction and examined its usefulness. CTP data from 13 patients with acute stroke were analyzed using the SVD and Bayesian estimation methods implemented in Vitrea. Evaluation of visual clarity of the ischemic area and quantitative values of the healthy side-affected side ratio using the mean values of the left and right region of interest (ROI) on the images were compared using the SVD and Bayesian estimation methods. In visual evaluation, there were significant differences in CBV in four cases, and in CBF, MTT, and TTP in many cases. The healthy side-affected side ratio of the SVD and Bayesian estimation methods were as follows: CBF 1.19, 1.84; CBV 1.09, 1.02; MTT 1.12, 1.79; and TTP 1.48, 1.19. For CBF and MTT, the Bayesian estimation method had a larger ratio of the healthy side to the affected side, and for TTP, the SVD method had a larger ratio of the test side to the affected side. We suggest that the Bayesian estimation method is more useful than the SVD method for assessing CBF and MTT in CTP analysis of patients with acute stroke.

ABSTRACTThis paper addresses the application of a Bayesian parameter estimation method to a regional seismic risk assessment of curved concrete bridges. For this purpose, numerical models of case-study bridges are simulated to generate multiparameter demand models of components, consisting of various uncertainty parameters and an intensity measure (IM). The demand models are constructed using a Bayesian parameter estimation method and combined with limit states to derive the parameterized fragility curves. These fragility curves are used to develop bridge-specific and bridge-class fragility curves. Moreover, a stepwise removal process in the Bayesian parameter estimation is performed to identify significant parameters affecting component demands.

This article compares maximum likelihood and Bayesian estimation of the correlated trait–correlated method (CT–CM) confirmatory factor model for multitrait–multimethod (MTMM) data. In particular, Bayesian estimation with minimally informative prior distributions—that is, prior distributions that prescribe equal probability across the known mathematical range of a parameter—are investigated as a source of information to aid convergence. Results from a simulation study indicate that Bayesian estimation with minimally informative priors produces admissible solutions more often maximum likelihood estimation (100.00% for Bayesian estimation, 49.82% for maximum likelihood). Extra convergence does not come at the cost of parameter accuracy; Bayesian parameter estimates showed comparable bias and better efficiency compared to maximum likelihood estimates. The results are echoed via 2 empirical examples. Hence, Bayesian estimation with minimally informative priors outperforms enables admissible solutions of the CT–CM model for MTMM data.

SummaryThe mixed logit model is widely used in applied econometrics. Researchers typically rely on the free choice between the classical and Bayesian estimation approach. However, empirical evidence of the similarity of their parameter estimates is sparse. The presumed similarity is mainly based on one empirical study that analyzes a single dataset (Huber J, Train KE. 2001. On the similarity of classical and Bayesian estimates of individual mean partworths. Marketing Letters12(3): 259–269). Our replication study offers a generalization of their results by comparing classical and Bayesian parameter estimates from six additional datasets and specifically for panel versus cross‐sectional data. In general, our results suggest that the two methods provide similar results, with less similarity for cross‐sectional data than for panel data. Copyright © 2016 John Wiley & Sons, Ltd.

This study examines the optimal selections of bandwidth and semi-metric for a functional partial linear model. Our proposed method begins by estimating the unknown error density using a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, can be estimated by functional principal component and functional Nadayara-Watson estimators. The estimation accuracy of the regression function and error density crucially depends on the optimal estimations of bandwidth and semi-metric. A Bayesian method is utilized to simultaneously estimate the bandwidths in the regression function and kernel error density by minimizing the Kullback-Leibler divergence. For estimating the regression function and error density, a series of simulation studies demonstrate that the functional partial linear model gives improved estimation and forecast accuracies compared with the functional principal component regression and functional nonparametric regression. Using a spectroscopy dataset, the functional partial linear model yields better forecast accuracy than some commonly used functional regression models. As a by-product of the Bayesian method, a pointwise prediction interval can be obtained, and marginal likelihood can be used to select the optimal semi-metric.

Forecasting ground-level irradiance over short horizons: Time series, meteorological, and time-varying parameter models

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.

The main purpose of this study is to deal with the parameter estimation problem for the geometric process (GP) when the distribution of the first occurrence time of an event is assumed to be Rayleigh. For this purpose, maximum likelihood and Bayesian parameter estimation methods are discussed. Lindley and MCMC approximation methods are used in Bayesian calculations. Additionally, a novel method called the Modified-Lindley approximation has been proposed as an alternative to the Lindley approximation. An extensive simulation study was conducted to compare the performances of the prediction methods. Finally, a real data set is analyzed for illustrative purposes.

Bayesian model selection and parameter estimation is attracting a lot of interest in the astronomical community because of its power and logical consistency. Markov chain Monte Carlo provides the computational power for Bayesian parameter estimation problems in large parameter spaces but needs to be supported with other numerical techniques for efficient exploration of multi-modal probability distributions. Bayesian model selection is easy in concept but remains a difficult challenge for large parameter spaces. My comments here on the paper by Roberto Trotta are based on lessons learned from developing a controlled statistical fusion approach to some of these issues.KeywordsMarkov Chain Monte CarloMarginal LikelihoodProposal DistributionMarkov Chain Monte Carlo AlgorithmMarkov Chain Monte Carlo SampleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is the second part of a three part article examining methods for Bayesian estimation and tracking. In the first part we presented the general theory of Bayesian estimation where we showed that Bayesian estimation methods can be divided into two very general classes: a class where the observation conditioned posterior densities are propagated in time through a predictor/corrector method; and a second class where the first two moments are propagated in time, with state and observation moment prediction steps followed by state moment update steps that use the latest observations. In this second part, we make the assumption that all densities are Gaussian and, after applying an affine transformation and approximating all nonlinear functions by interpolating polynomials, we recover the sigma‐point class of Kalman filters, including the unscented, spherical simplex, and Gauss‐Hermite Kalman filters. In part 3, we will show that approximating a density by a set of Monte Carlo samples leads to particle filter methods, where the posterior density is propagated in time and moment integrals are approximated by sample moments. WIREs Comput Stat 2012 doi: 10.1002/wics.1215This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory

Bayesian point estimation is commonly used for system identification owing to its good properties for small sample sizes. Although this type of estimator is usually non-parametric, Bayes estimates can also be obtained for rational parametric models, which is often of interest. However, as in maximum-likelihood methods, the Bayes estimate is typically computed via local numerical optimization that requires good initialization and cannot guarantee global optimality. In this contribution, we propose a computationally tractable method that computes the Bayesian parameter estimates with posterior certification of global optimality via sum-of-squares polynomials and sparse semidefinite relaxations. It is shown that the method is applicable to certain discrete-time linear models, which takes advantage of the rational structure of these models and the sparsity in the Bayesian parameter estimation problem. The method is illustrated on a simulation model of a resonant system that is difficult to handle when the sample size is small.

Biological processes underlying skin cancer growth and wound healing are governed by various collective cell spreading mechanisms. This thesis develops new statistical methods to provide key insights into the mechanisms driving the spread of cell populations such as motility, proliferation and cell-to-cell adhesion, using experimental data. The new methods allow us to precisely estimate the parameters of such mechanisms, quantify the associated uncertainty and investigate how these mechanisms are influenced by various factors. The thesis provides a useful tool to measure the efficacy of medical treatments that aim to influence the spread of cell populations.

BackgroundTranslating a known metabolic network into a dynamic model requires reasonable guesses of all enzyme parameters. In Bayesian parameter estimation, model parameters are described by a posterior probability distribution, which scores the potential parameter sets, showing how well each of them agrees with the data and with the prior assumptions made.ResultsWe compute posterior distributions of kinetic parameters within a Bayesian framework, based on integration of kinetic, thermodynamic, metabolic, and proteomic data. The structure of the metabolic system (i.e., stoichiometries and enzyme regulation) needs to be known, and the reactions are modelled by convenience kinetics with thermodynamically independent parameters. The parameter posterior is computed in two separate steps: a first posterior summarises the available data on enzyme kinetic parameters; an improved second posterior is obtained by integrating metabolic fluxes, concentrations, and enzyme concentrations for one or more steady states. The data can be heterogenous, incomplete, and uncertain, and the posterior is approximated by a multivariate log-normal distribution. We apply the method to a model of the threonine synthesis pathway: the integration of metabolic data has little effect on the marginal posterior distributions of individual model parameters. Nevertheless, it leads to strong correlations between the parameters in the joint posterior distribution, which greatly improve the model predictions by the following Monte-Carlo simulations.ConclusionWe present a standardised method to translate metabolic networks into dynamic models. To determine the model parameters, evidence from various experimental data is combined and weighted using Bayesian parameter estimation. The resulting posterior parameter distribution describes a statistical ensemble of parameter sets; the parameter variances and correlations can account for missing knowledge, measurement uncertainties, or biological variability. The posterior distribution can be used to sample model instances and to obtain probabilistic statements about the model's dynamic behaviour.

The chapter “Bayesian Statistics” gives a brief overview of the Bayesian approach to statistical analysis. It starts off by examining the difference between frequentist statistics and Bayesian statistics. Next, it introduces Bayes’ theorem and explains how the theorem is used in statistics and model selection, with the prosecutor’s fallacy given as a practice example. The chapter then goes on to discuss priors and Bayesian parameter estimation. It concludes with some final thoughts on Bayesian approaches. The chapter does not answer the question “Should ecologists become Bayesian?” However, to the extent that alternative models can be posed as alternative values of parameters, Bayesian parameter estimation can help assign probabilities to those hypotheses.