Linear Bayes Methods Research Articles

Bayes estimator for the inequality-constrained regression model*

This article studies the Bayesian estimation of the inequality-constrained regression model under a balanced loss function. The usual Bayes estimator is often involved in some complex integrals and has no explicit expression, which causes the corresponding properties be difficult to describe. Based on the idea of linear Bayes method suggested by Rao (1973), the present paper constructs a linear Bayes estimator for the regression parameters without specify their specific prior distributions. It is proved that the proposed linear Bayes estimator outperforms the corresponding least squares estimator, and numerical simulations show that when there is collinearity in design matrix, the linear Bayes estimator performs better than the usual Bayes estimator.

Bayesian instrumental variable estimation in linear measurement error models

AbstractIn this article, we study the problem of parameter estimation for measurement error models by combining the Bayes method with the instrumental variable approach, deriving the posterior distribution of parameters under different priors with known and unknown variance parameters, respectively, and calculating the Bayes estimator (BE) of the parameters under quadratic loss. However, it is difficult to obtain an explicit expression for BE because of the complex multiple integrals involved. Therefore, we adopt the linear Bayes method, which does not specify the form of the prior and avoids these complicated integral calculations, to obtain an expression for the linear Bayes estimator (LBE) for different priors. We prove that this LBE is superior to the two‐stage least squares estimator under the mean squared error matrix criterion. Numerical simulations show that our LBE is very close to the real parameter whether the variance parameters are known or unknown, and it gradually approaches BE as the sample size increases. Our results indicate that this instrumental variable approach is valid for measurement error models.

Estimation of structural parameters in balanced Bühlmann credibility model with correlation risk

In this paper, the longitudinal data analysis is used to interpret the balanced Bühlmann credibility model with correlation risk, and the homogeneous credibility estimator is derived. We obtain the restricted maximum likelihood estimators (RMLE) for the structural parameters involved in the credibility factor and show that they are unbiased. In addition, the linear Bayes method is employed to estimate the structural parameters, and the proposed linear Bayes estimators (LBE) appear to outperform RMLE in terms of the mean squared error matrix (MSEM) criterion. Simulation studies show that the proposed LBE performs well.

Dynamic Mixture of Experts Models for Online Prediction

A mixture of experts models the conditional density of a response variable using a mixture of regression models with covariate-dependent mixture weights. We extend the finite mixture of experts model by allowing the parameters in both the mixture components and the weights to evolve in time by following random walk processes. Inference for time-varying parameters in richly parameterized mixture of experts models is challenging. We propose a sequential Monte Carlo algorithm for online inference based on a tailored proposal distribution built on ideas from linear Bayes methods and the EM algorithm. The method gives a unified treatment for mixtures with time-varying parameters, including the special case of static parameters. We assess the properties of the method on simulated data and on industrial data where the aim is to predict software faults in a continuously upgraded large-scale software project.

Linear Bayes estimator of the extreme value distribution based on type II censored samples

In this paper, a linear Bayes method is employed to simultaneously estimate the location parameter and the scale parameter of the extreme value distribution. Based on type II censored samples, we construct the linear Bayes estimator (LBE) of the parameter vector and establish its superiority over the classical unbiased estimator in terms of the mean square error matrix criterion. The proposed LBE is easy to calculate, and numerical results are presented to verify that the LBE works well. The estimator is further applied to a real data case to demonstrate its feasibility.

On estimating uniform distribution via linear Bayes method

ABSTRACTA linear Bayes procedure is suggested to simultaneously estimate the parameters of the uniform distribution U(θ1, θ2). The proposed linear Bayes estimator is simple and easy to use and its superiorities are established.

How to Improve Classical Estimators via Linear Bayes Method?

In this survey, we use the normal linear model to demonstrate the use of the linear Bayes method. The superiorities of linear Bayes estimator (LBE) over the classical UMVUE and MLE are established in terms of the mean squared error matrix (MSEM) criterion. Compared with the usual Bayes estimator (obtained by the MCMC method) the proposed LBE is simple and easy to use with numerical results presented to illustrate its performance. We also examine the applications of linear Bayes method to some other distributions including two-parameter exponential family, uniform distribution and inverse Gaussian distribution, and finally make some remarks.

Credibility Estimation of Distribution Functions with Applications to Experience Rating in General Insurance

This article presents a new credibility estimation of the probability distributions of risks under Bayes settings in a completely nonparametric framework. In contrast to the Ferguson's Bayesian nonparametric method, it does not need to specify a mathematical form of the prior distribution (such as a Dirichlet process). We then show the applications of the method in general insurance premium pricing, a procedure commonly known as experience rating, which utilizes the insured's claim experience to calculate a proper premium under a given premium principle (referred to as a risk measure). As this method estimates the probability distributions of losses, not just the means and variances, it provides a unified nonparametric framework to experience rating for arbitrary premium principles. This encompasses the advantages of the well-known Bühlmann's and Ferguson's approaches, while it overcomes their drawbacks. We first establish a linear Bayes method and prove its strong consistency in nonparametric settings that require only knowledge of the first two moments of the loss distributions considered as a stochastic process. Then an empirical Bayes method is developed for the more general situation where a portfolio of risks is observed but no knowledge is available or assumed on their loss and prior distributions, including their moments. It is shown to be asymptotically optimal. The performance of our estimates in comparison with traditional methods is also evaluated through theoretical analysis and numerical studies, which show that our approach produces premium estimates close to the optima.

Abstract 3995: Breast cancer molecular subtypes are reflected in circulating microRNA profiles

Abstract Background: Breast cancer is a heterogeneous disease. Triple-negative breast cancer (TNBC) as defined by negativity in estrogen receptor (ER), progesterone receptor (PR) and human epidermal growth factor receptor 2 (HER2), has a distinct disease course and response to treatment, compared with ER+ breast cancer. Circulating microRNAs (miRNAs) have been suggested as promising diagnostic and early detection biomarkers for cancer, including breast cancer, as miRNAs can be detected in extra-cellular biofluids at stable levels. However, findings from previous studies of circulating miRNA are inconsistent, which may be in part due to the heterogeneity nature of breast cancer. Methods: To characterize the differential expression pattern of circulating miRNAs in TNBC from ER+ breast cancer, we performed a whole miRNome profiling experiment using Exiqon miRCURY microRNA qRT-PCR Human panels which include 752 miRNAs. Total RNA was extracted from 200ul serum of 27 luminal A-like breast cancer patients (ER+/PR+/HER2-), 18 TNBC, and 31 volunteer controls who did not have breast cancer. After global mean normalization, we used linear models and empirical Bayes methods to examine expression differences across the three groups. Benjamini and Hochberg's false discovery rate method was used to correct for multiple testing. Penalized logistic regression was used to build a profile of miRNAs that can distinguish TNBC from luminal A-like breast cancer. Results: The mean age was similar between the three groups (luminal A-like: 51.1, TNBC: 52.5, and controls: 50.6 years old). A total of 226 miRNAs could be detected in more than half of the samples and were included for further analysis. Of them, 20 miRNAs were differently expressed between luminal A-like and control groups, and 12 were different between TNBC and controls. In the contrast between TNBC and luminal A-like breast cancer, we found 34 miRNAs were significantly different, with the most significant miRNA being miR-423 (p = 8.8e-5). The multivariable logistic regression identified a panel of 7 miRNAs which can distinguish TNBC and luminal A-like tumors. ROC analysis found these 7 miRNAs had good discriminatory power (area under ROC curve = 0.865). Conclusion: Our study demonstrates the existence of a set of circulating miRNAs specific to TNBC, which has a clear “fingerprint” to differentiate TNBC from luminal A-like breast cancer. This finding suggests that early detection markers for different breast cancer subtypes may be distinct. It is also interesting to further investigate if different miRNA profiles in circulation between breast cancer subtypes are derived directly from tumors or from other sources in response to tumors. Citation Format: Dezheng Huo, Wendy M. Clayton, Toshio F. Yoshimatsu, Jianjun Chen, Olufunmilayo I. Olopade. Breast cancer molecular subtypes are reflected in circulating microRNA profiles. [abstract]. In: Proceedings of the 106th Annual Meeting of the American Association for Cancer Research; 2015 Apr 18-22; Philadelphia, PA. Philadelphia (PA): AACR; Cancer Res 2015;75(15 Suppl):Abstract nr 3995. doi:10.1158/1538-7445.AM2015-3995

Bayesian Computational Tools

This article surveys advances in the field of Bayesian computation over the past 20 years from a purely personal viewpoint, hence containing some ommissions given the spectrum of the field. Monte Carlo, MCMC, and ABC themes are covered here, whereas the rapidly expanding area of particle methods is only briefly mentioned and different approximative techniques such as variational Bayes and linear Bayes methods do not appear at all. This article also contains some novel computational entries on the double-exponential model that may be of interest.

Linear Bayes estimator for the two-parameter exponential family under type II censoring

For the two-parameter exponential family, a linear Bayes method is proposed to simultaneously estimate the parameter vector consisting of location and scale parameters. The superiority of the proposed linear Bayes estimator (LBE) over the classical UMVUE is established in terms of the mean square error matrix (MSEM) criterion. The proposed LBE is simple and easy to use compared with the usual Bayes estimator, which is obtained by the MCMC method. Numerical results are presented to verify that the LBE works well. In the empirical Bayes framework, the paper invokes a linear empirical Bayes estimator (LEBE) by using a linear combination of historical samples. It is shown under some mild regularity conditions that the LEBE is superior to the classical UMVUE and the maximum likelihood estimator in terms of MSEM. It is further shown with numerical results that the performance of LEBE gets better with the increase in the number of historical samples.

A Bayes Linear Bayes Method for Estimation of Correlated Event Rates

Typically, full Bayesian estimation of correlated event rates can be computationally challenging since estimators are intractable. When estimation of event rates represents one activity within a larger modeling process, there is an incentive to develop more efficient inference than provided by a full Bayesian model. We develop a new subjective inference method for correlated event rates based on a Bayes linear Bayes model under the assumption that events are generated from a homogeneous Poisson process. To reduce the elicitation burden we introduce homogenization factors to the model and, as an alternative to a subjective prior, an empirical method using the method of moments is developed. Inference under the new method is compared against estimates obtained under a full Bayesian model, which takes a multivariate gamma prior, where the predictive and posterior distributions are derived in terms of well-known functions. The mathematical properties of both models are presented. A simulation study shows that the Bayes linear Bayes inference method and the full Bayesian model provide equally reliable estimates. An illustrative example, motivated by a problem of estimating correlated event rates across different users in a simple supply chain, shows how ignoring the correlation leads to biased estimation of event rates.

Inference of Dynamic Generalized Linear Models: On‐Line Computation and Appraisal

Summary The purpose of this paper is to provide a critical discussion on real‐time estimation of dynamic generalized linear models. We describe and contrast three estimation schemes, the first of which is based on conjugate analysis and linear Bayes methods, the second based on posterior mode estimation, and the third based on sequential Monte Carlo sampling methods, also known as particle filters. For the first scheme, we give a summary of inference components, such as prior/posterior and forecast densities, for the most common response distributions. Considering data of arrivals of tourists in Cyprus, we illustrate the Poisson model, providing a comparative analysis of the above three schemes.

Extended Bayesian Inference Method for Evaluating Pipe Failure Probability in Corrosion Rate Fluctuation Model

Because of the lessening number of maintenance experts, a method to rationalize pipe inspection interval is desired. For this purpose, the evaluation method for the pipe integrity in the form of failure probability has been developed based on the Bayesian inference method in the previous paper. In this paper, the previous method is called the linear-Bayes method. The linear-Bayes method assumes wall thinning due to Flow Accelerated Corrosion (FAC) as the principal damage mechanism, and it can define the safety margin of a pipe's residual life depending on the number of inspections. However, the linear-Bayes method ignores the corrosion rate fluctuation against time, which may be caused by the change of environment such as water chemistry and flow velocity. Therefore, the linear-Bayes method may underestimate the failure probability of the pipe segments if the online monitoring of the environments is not used. In this paper, the linear-Bayes method is extended for the wall-thinning model with the corrosion rate fluctuation. The extension is carried out through following two approaches : correction-term and error-term approaches. In this paper, the formulation and the procedure for each approach are shown at first. And then, the accuracy and the merit of the extended method are examined through the evaluation using the artificial and the actual inspection records. Through the examination, it is confirmed that the extended method makes the evaluation of the pipe integrity available in view of safety margin for the corrosion rate fluctuation, keeping the merits of the linear-Bayes method.