Bayesian Bootstrap Method Research Articles

On the robustness of thermal comfort against uncertain future climate: A Bayesian bootstrap method

Climate change mitigation and adaptation warrants their synergetic consideration in the building design process, yet past decades have witnessed an unbalanced focus on the mitigation of energy and carbon. In redressing the imbalance, the major challenge lies in the accurate prediction of future building performance via building energy modelling, which is considerably hindered by uncertainties in future climate data. Robustness analysis is a promising technique to inform uncertainty-based decision-making, but its application to future thermal comfort has yet to be sufficiently explored in the built environment. From the perspective of domestic overheating, this paper represents an initial investigation into the implementation of the Bayesian bootstrap method, to quantitatively evaluate the robustness of thermal comfort against uncertain future climate. This is demonstrated using a case study of two typical post-retrofit dwellings in England, where the Bayesian bootstrap also enables the statistical comparison of their expected future overheating risk with climate uncertainty considered. The main findings reveal the magnitude of both overheating risk and its variability experienced during nocturnal occupancy in regulation-compliant dwellings, respectively comprising nearly 15 and 12 times greater than during daytime in extreme cases. Results also imply that adaptive ventilation is potentially the key measure to enhance the robustness of thermal comfort against climate uncertainty. Overall, the Bayesian bootstrap is shown to provide a systematically consistent approach to the robustness assessment of future thermal comfort, which can facilitate the comparability of design alternatives that is vital to the building design decision-making process integrating both mitigation and adaptation strategies.

Evaluating severity–area–frequency (SAF) of seasonal droughts in Bangladesh under climate change scenarios

Drought is considered to be one of the most devastating natural hazards, causing widespread environmental and social damage in many parts of the world. Using standardized precipitation index, this work has assessed changes in the severity–area–frequency (SAF) relationship curve of seasonal droughts in Bangladesh. Changes were estimated for mild, moderate, severe and extreme droughts for the four climatic seasons; winter, pre-monsoon, monsoon, post-monsoon, and for the two major growing seasons; rabi (November to April) and kharif (May to October). Nineteen general circulation models (GCMs) of Couple Model Intercomparison Project 5 were used. The model output statistics approach was used to downscale GCM simulated rainfall for eighteen climate stations in Bangladesh. Changes in the SAF curve were computed for three periods (2010–2039, 2040–2069 and 2070–2099). The uncertainty band of the SAF relationship curve was then computed using the Bayesian bootstrap method at the 95% confidence level. The results reveal that moderate and severe drought categories have the highest return period and are likely to affect the region more than other types of droughts. The kharif season drought was found to be most pronounced and affected significant portions of the country during all return periods and severity categories. Projections also show that monsoon and kharif droughts would increase across Bangladesh in regards of severity and return period. Higher return period droughts were also projected to increase in aerial extent in the middle of this century (2040–2069).

Determination of statistical discontinuity persistence for a rock mass characterized by non-persistent fractures

The discontinuity persistence is an important parameter that affects the strength of rock masses and stability in rock engineering. However, determining the 3D discontinuity persistence in practical field projects remains difficult. This study proposes an approach to determine the 3D discontinuity persistence (percentage of total discontinuity area to the area of the potential failure surface) on the basis of Dijkstra's shortest-path algorithm and Bayesian bootstrap method. First, Dijkstra's shortest-path algorithm is used to determine the linear discontinuity persistence (percentage of total discontinuity trace length to the length of 2D potential failure surface), which is essential to determining 3D discontinuity persistence. Second, geological analysis is performed to deduce 3D discontinuity persistence as a function related to two linear discontinuity persistence in two vertical directions. Lastly, Bayesian bootstrap method is used to analyze the linear and 3D discontinuity persistence. This approach can substantially simplify the process of determining discontinuity persistence and comprehensively considers the uncertainty of the fracture network. The mean, standard deviation, and p-value (i.e., value at the 95% confidence limit) are used as the statistical parameters of the linear and 3D discontinuity persistence.

Bayesian estimation of Weibull mixture in heavily censored data setting

In reliability or warranty analysis, engineers must often deal with lifetimes data that are non-homogeneous. Most of the time, this variability is unobserved but has to be taken into account for reliability or warranty cost analysis. A further problem is that in reliability analysis, data are heavily censored which makes estimations more difficult. The two-component Weibull mixture is then a highly relevant model to capture heterogeneity for a large majority of operating lifetimes. Unfortunately, the performance of classical estimation methods (maximum of likelihood via EM, Bayes approach via MCMC) is jeopardized due to the high number of parameters and the heavy censoring. In order to overcome the problem of heavy censoring for Weibull mixture parameters estimation, this research proposes a Bayesian bootstrap method, called Bayesian Restoration Maximization. The key is to provide a sampling from the posterior distribution. Thanks to an importance sampling technique, this sample focuses on the posterior mean. Prior distributions elicitation and sensibility analysis are discussed. Simulations results showed that, for heavily censored data, the BRM method outperforms the EM and S-EM algorithms in terms of estimates accuracy. On the other hand, it is a non-iterative method which therefore provides very short computation times. In addition, the BRM method does not suffer from the problem of label switching from Bayesian sampling algorithms such as MCMC methods. Finally, two real data sets are analyzed to illustrate the application of the method.

ESTIMATION OF THE BINARY LOGISTIC REGRESSION MODEL PARAMETER USING BOOTSTRAP RE-SAMPLING

In this paper, the non-parametric bootstrap and non-parametric Bayesian bootstrap methods are applied for parameter estimation in the binary logistic regression model. A real data study and a simulation study are conducted to compare the Nonparametric bootstrap, Non-parametric Bayesian bootstrap and the maximum likelihood methods. Study results shows that three methods are all effective ways for parameter estimation in the binary logistic regression model. In small sample case, the non-parametric Bayesian bootstrap method performs relatively better than the non-parametric bootstrap and the maximum likelihood method for parameter estimation in the binary logistic regression model.

A two-step semiparametric method to accommodate sampling weights in multiple imputation.

Multiple imputation (MI) is a well-established method to handle item-nonresponse in sample surveys. Survey data obtained from complex sampling designs often involve features that include unequal probability of selection. MI requires imputation to be congenial, that is, for the imputations to come from a Bayesian predictive distribution and for the observed and complete data estimator to equal the posterior mean given the observed or complete data, and similarly for the observed and complete variance estimator to equal the posterior variance given the observed or complete data; more colloquially, the analyst and imputer make similar modeling assumptions. Yet multiply imputed data sets from complex sample designs with unequal sampling weights are typically imputed under simple random sampling assumptions and then analyzed using methods that account for the sampling weights. This is a setting in which the analyst assumes more than the imputer, which can led to biased estimates and anti-conservative inference. Less commonly used alternatives such as including case weights as predictors in the imputation model typically require interaction terms for more complex estimators such as regression coefficients, and can be vulnerable to model misspecification and difficult to implement. We develop a simple two-step MI framework that accounts for sampling weights using a weighted finite population Bayesian bootstrap method to validly impute the whole population (including item nonresponse) from the observed data. In the second step, having generated posterior predictive distributions of the entire population, we use standard IID imputation to handle the item nonresponse. Simulation results show that the proposed method has good frequentist properties and is robust to model misspecification compared to alternative approaches. We apply the proposed method to accommodate missing data in the Behavioral Risk Factor Surveillance System when estimating means and parameters of regression models.

Statistical Analysis of Aircraft–Bird Strikes Resulting in Engine Failure

Engine failure caused by bird strikes can be particularly perilous for today's typical twin-engine aircraft. Although large-bird populations have increased substantially since the 1970s, modern-day turbofan engines are not tested for large birds. Instead, it is acceptable for contemporary turbofan engines to lose all power because of large-bird ingestion. With the increasing use of turbofan engines and air traffic, not only are more bird strikes expected in the near future, but also more bird strikes are anticipated to result in engine failure. This study identified the factors that were statistically associated with the probability of engine failure in the event of a bird strike. A large sample of more than 42,000 U.S. bird strikes was used. The missing data in the sample were multiply imputed by using an approximate Bayesian bootstrap method. With the multiply imputed data, 15 factors were statistically analyzed. Six of those factors were found to be significantly associated with the probability of engine failure in the event of a bird strike: altitude above ground level, bird size, number of birds struck, flight phase, daylight conditions, and sky conditions. A logistic regression model was developed, and a detailed probabilistic interpretation of the model is given for practitioners. By using the findings, aviation authorities can improve bird strike hazard mitigation strategies, flight crews can reduce the potential of bird strikes resulting in engine failure, and researchers can better understand the nature of bird strikes and develop a scientific approach to minimize the likelihood of engine failure in the event of a bird strike.

Bayesian Monitoring and Bootstrap Trial Simulation: A New Paradigm to Implement Adaptive Trial Design for Testing Antidepressant Drugs

A novel methodology is proposed for continuous monitoring of efficacy data in ongoing antidepressant clinical trials and for decision making to support progression or discontinuation of the trial or one of the treatment arms. The Posterior Probability of Superiority (PPS) resulting from the application of Monte Carlo Markov Chain approach to a longitudinal model describing the time course of placebo and antidepressant drugs was used to estimate criteria to discontinue a treatment arm or the trial for futility, and to predict the treatment effect at study-end while the trial was still ongoing. The decision to stop the study was based on PPS, Predictive Power and on risk analysis based on a non-parametric bootstrap trial simulation. The performance of the Bayesian monitoring was evaluated by the retrospective analysis of 3 clinical trials. The Bootstrap-based methodology was compared to the Conditional Power and the Predictive Probability approaches. The application of the proposed methodology showed the possibility to stop a trial for futility when about 50% of total information was available and to detect signal of a treatment effect when limited information (<40%) was available. The comparisons with the Condition Power and the Predictive Probability approaches indicated that the Bayesian Bootstrap method, based on data-driven assumptions for priors, provided a better control for the risk of inappropriate decisions. The results suggest that the proposed methodology to monitor the accumulating information and to provide a scenario- based risk analysis could constitute a valuable approach to re-engineer the development process of novel drugs.

The bootstrap and Bayesian bootstrap method in assessing bioequivalence

Parametric method for assessing individual bioequivalence (IBE) may concentrate on the hypothesis that the PK responses are normal. Nonparametric method for evaluating IBE would be bootstrap method. In 2001, the United States Food and Drug Administration (FDA) proposed a draft guidance. The purpose of this article is to evaluate the IBE between test drug and reference drug by bootstrap and Bayesian bootstrap method. We study the power of bootstrap test procedures and the parametric test procedures in FDA (2001). We find that the Bayesian bootstrap method is the most excellent.

Bayesian bootstrap estimation of ROC curve

Receiver operating characteristic (ROC) curve is widely applied in measuring discriminatory ability of diagnostic or prognostic tests. This makes the ROC analysis one of the most active research areas in medical statistics. Many parametric and semiparametric estimation methods have been proposed for estimating the ROC curve and its functionals. In this paper, we propose the Bayesian bootstrap (BB), a fully nonparametric estimation method, for the ROC curve and its functionals, such as the area under the curve (AUC). The BB method offers a bandwidth-free smoothing approach to the empirical estimate, and gives credible bounds. The accuracy of the estimate of the ROC curve in the simulation studies is examined by the integrated absolute error. In comparison with other existing curve estimation methods, the BB method performs well in terms of accuracy, robustness and simplicity. We also propose a procedure based on the BB approach to test the binormality assumption.

Strong approximations for resample quantile processes and application to ROC methodology

The receiver operating characteristic (ROC) curve is defined as true positive rate versus false positive rate obtained by varying a decision threshold criterion. It has been widely used in medical sciences for its ability to measure the accuracy of diagnostic or prognostic tests. Mathematically speaking, ROC curve is the composition of survival function of one population with the quantile function of another population. In this paper, we study strong approximation for the quantile processes of the Bayesian bootstrap (BB) resampling distributions, and use this result to study strong approximations for the BB version of the ROC process in terms of two independent Kiefer processes. The results imply asymptotically accurate coverage probabilities for the confidence bands for the ROC curve and confidence intervals for the area under the curve functional of the ROC constructed using the BB method. Similar results follow for the bootstrap resampling distribution.

Case Based Imprecision Estimates for Bayes Classifiers with the Bayesian Bootstrap

This article outlines a Bayesian bootstrap method for case based imprecision estimates in Bayes classification. We argue that this approach is an important complement to methods such as k-fold cross validation that are based on overall error rates. It is shown how case based imprecision estimates may be used to improve Bayes classifiers under asymmetrical loss functions. In addition, other approaches to making use of case based imprecision estimates are discussed and illustrated on two real world data sets. Contrary to the common assumption, Bayesian bootstrap simulations indicate that the uncertainty associated with the output of a Bayes classifier is often far from normally distributed.

An approximate Bayesian risk-analysis for the gastro-intestinal safety of ibuprofen.

Although several studies on ibuprofen and its gastro-intestinal (GI) risk have been reported, the dose-response relationship was not clear due to the lack of information regarding high-dose exposure. Analysis using Bayesian methods is appropriate whenever data are sparse, although such methods are not easily implemented. A retrospective cohort study to assess this dose-response relationship was carried out using a record linkage database. A Bayesian risk-analysis was conducted using the Bayesian bootstrap approximation. Risks of GI events at different dose levels were compared using the posterior distributions and the number of events predicted to occur in the future was estimated. Risk factors such as age, gender and co-morbidity were adjusted for in the analysis. This approximation was compared with the full Bayesian approach using the usual but more computer-intensive tool of Markov Chain Monte Carlo simulation. There were 1, 5 and 10 complicated GI events during exposure to high, medium and low dose ibuprofen with 0.2, 1.8 and 7.0 thousand person-years (PY) exposure, respectively. After adjusting for other risk factors the relative risks of high versus low and medium versus low doses were 6.3 (95% CI = 0.21, 24.17) and 2.5 (95% CI = 0.71, 5.85), respectively. Using the approximate Bayesian method prediction of the number of events in a population of females aged 50-59 with no previous medical problems with 1000 PY drug exposure showed that the estimated probability of having more than five events was 0.048 for the medium-dose group and 0.14 for the high-dose group. High dose ibuprofen appears to have a considerably greater risk of having a larger number of adverse GI events than a medium dose. The approximate Bayesian bootstrap method was demonstrated to be a robust and easily implemented alternative to the full Bayesian approach to risk analysis whenever data are sparse.

A note on approximate Bayesian bootstrap imputation

The approximate Bayesian bootstrap is suggested by Rubin & Schenker (1986) as a way of generating multiple imputations when the original sample can be regarded as independently and identically distributed and the response mechanism is ignorable. We investigate the finite sample properties of the variance estimator when the approximate Bayesian bootstrap method is used and show that the bias is not negligible for moderate sample sizes. A modification of the method is proposed for reducing the bias of the variance estimator. The proposed method is asymptotically equivalent to the approximate Bayesian bootstrap method but has better finite sample properties.

Convergence rates for density estimation with Bernstein polynomials

Mixture models for density estimation provide a very useful set up for the Bayesian or the maximum likelihood approach.For a density on the unit interval, mixtures of beta densities form a flexible model. The class of Bernstein densities is a much smaller subclass of the beta mixtures defined by Bernstein polynomials, which can approximate any continuous density. A Bernstein polynomial prior is obtained by putting a prior distribution on the class of Bernstein densities. The posterior distribution of a Bernstein polynomial prior is consistent under very general conditions. In this article, we present some results on the rate of convergence of the posterior distribution. If the underlying distribution generating the data is itself a Bernstein density, then we show that the posterior distribution converges at “nearly parametric rate” $(log n) /\sqrt{n}$ for the Hellinger distance. If the true density is not of the Bernstein type, we show that the posterior converges at a rate $n^{1/3}(log n)^{5/6}$ provided that the true density is twice differentiable and bounded away from 0. Similar rates are also obtained for sieve maximum likelihood estimates.These rates are inferior to the pointwise convergence rate of a kernel type estimator.We show that the Bayesian bootstrap method gives a proxy for the posterior distribution and has a convergence rate at par with that of the kernel estimator.

Experience with a bayesian bootstrap method incorporating proper prior information

Although bootstrapping has become widely used in statistical analysis, there has been little reported concerning bootstrapped Bayesian analyses, especially when there is proper prior informa-tion concerning the parameter of interest. In this paper, we first propose an operationally implementable definition of a Bayesian bootstrap. Thereafter, in simulated studies of the estimation of means and variances, this Bayesian bootstrap is compared to various parametric procedures. It turns out that little information is lost in using the Bayesian bootstrap even when the sampling distribution is known. On the other hand, the parametric procedures are at times very sensitive to incorrectly specified sampling distributions, implying that the Bayesian bootstrap is a very robust procedure for determining the posterior distribution of the parameter.