Selection Of Prior Distributions Research Articles

Enhancing Study Power in Pediatric Research through Bayesian Extrapolation

Pediatric populations, comprising individuals aged 18 or younger, and rare disease populations pose significant challenges in clinical trial design due to limited participant numbers, patient sensitivity, and insufficient natural history data. Addressing these challenges often involves data extrapolation, leveraging existing data from adults to inform pediatric trials. Bayesian hierarchical modeling is increasingly recognized as a valuable tool for combining information across disparate sources, such as adult and pediatric datasets. This manuscript introduces an extension of an existing statistical model to enhance the efficiency of borrowing strength from multiple historical trials under consistent assumptions. A quantitative method is developed to improve the borrowing of historical information. The methods are illustrated with a simulation study motivated by real case study in pediatric clinical trial, and practical considerations are provided regarding the selection of prior distributions. This work aims to provide comprehensive insights and practical guidance for leveraging historical data effectively in clinical trial design.

A Comparison of Methods for Synthesizing Results from Previous Research to Obtain Priors for Bayesian Structural Equation Modeling

Bayesian estimation of latent variable models provides some unique advantages to researchers working with small samples and complex models when compared with the more commonly used maximum likelihood approach. A key aspect of Bayesian modeling involves the selection of prior distributions for the parameters of interest. Prior research has demonstrated that using default priors, which are typically noninformative, may yield biased and inefficient estimates. Therefore, it is recommended that data analysts obtain useful, informative priors from prior research whenever possible. The goal of the current simulation study was to compare several methods designed to combine results from prior studies that will yield informative priors for regression coefficients in structural equation models. These methods include noninformative priors, Bayesian synthesis, pooled analysis, aggregated priors, standard meta-analysis, power priors, and the meta-analytic predictive methods. Results demonstrated that power priors and meta-analytic predictive priors, used in conjunction with Bayesian estimation, may yield the most accurate estimates of the latent structure coefficients. Implications for practice and suggestions for future research are discussed.

Testing Data Cloning as the Basis of an Estimator for the Stochastic Volatility in Mean Model

Developed as a refinement of stochastic volatility (SV) models, the stochastic volatility in mean (SVM) model incorporates the latent volatility as an explanatory variable in both the mean and variance equations. It, therefore, provides a way of assessing the relationship between returns and volatility, albeit at the expense of complicating the estimation process. This study introduces a Bayesian methodology that leverages data-cloning algorithms to obtain maximum likelihood estimates for SV and SVM model parameters. Adopting this Bayesian framework allows approximate maximum likelihood estimates to be attained without the need to maximize pseudo likelihood functions. The key contribution this paper makes is that it proposes an estimator for the SVM model, one that uses Bayesian algorithms to approximate the maximum likelihood estimate with great effect. Notably, the estimates it provides yield superior outcomes than those derived from the Markov chain Monte Carlo (MCMC) method in terms of standard errors, all while being unaffected by the selection of prior distributions.

Estimation of Ordinary Differential Equation Models for Gene Regulatory Networks Through Data Cloning.

Ordinary differential equations (ODEs) are widely used for elucidating dynamic processes in various fields. One of the applications of ODEs is to describe dynamics of gene regulatory networks (GRNs), which is a critical step in understanding disease mechanisms. However, estimation of ODE models for GRNs is challenging because of inflexibility of the model and noisy data with complex error structures such as heteroscedasticity, correlations between genes, and time dependency. In addition, either a likelihood or Bayesian approach is commonly used for estimation of ODE models, but both approaches have benefits and drawbacks in their own right. Data cloning is a maximum likelihood (ML) estimation method through the Bayesian framework. Since it works in the Bayesian framework, it is free from local optimum problems that are common drawbacks of ML methods. Also, its inference is invariant for the selection of prior distributions, which is a major issue in Bayesian methods. This study proposes an estimation method of ODE models for GRNs through data cloning. The proposed method is demonstrated through simulation and it is applied to real gene expression time-course data.

Short-Term Eruption Forecasting for Crisis Decision-Support in the Auckland Volcanic Field, New Zealand

Auckland, a city of 1.6 million people, is situated atop the active monogenetic Auckland Volcanic Field (AVF). Thus, short-term eruption forecasting is critical to support crisis management in a future event, especially to inform decisions such as calling evacuations. Here we present an updated BET_EF for the AVF incorporating new data and the results of an expert-opinion workshop, and test the performance of the resulting BETEF_AVF on eight hypothetical eruption scenarios with pre-eruptive sequences. We carry out a sensitivity analysis into the selection of prior distributions for key model parameters to explore the utility of using BET_EF outputs as a potential input for evacuation decision making in areas of distributed volcanism such as the AVF. BETEF_AVF performed well based on the synthetic unrest dataset for assessing the probability of eruption, with the vent outbreaks eventuating within the zone of high spatial likelihood. Our analysis found that the selection of different spatial prior model inputs affects the estimated vent location due to the weighting between prior models and monitoring inputs within the BET_EF, which as unrest escalates may not be appropriate for distributed volcanic fields. This issue is compounded when the outputs are combined with cost-benefit analysis to inform evacuation decisions, leading to areas well beyond those with observed precursory activity being included in evacuation zones. We find that several default settings used in past work for the application of BET_EF and CBA to inform evacuation decision-support are not suitable for distributed volcanism; in particular, the default 50-50 weighting between priors and monitoring inputs for assessing spatial vent location does not produce useful results. We conclude by suggesting future cost-benefit analysis applications in volcanic fields appropriately consider the spatial and temporal variability and uncertainty characteristic of such systems.

Parameter estimation of multivariate multiple regression model using bayesian with non-informative Jeffreys’ prior distribution

Bayesian method is a method that can be used to estimate the parameters of multivariate multiple regression model. Bayesian method has two distributions, there are prior and posterior distributions. Posterior distribution is influenced by the selection of prior distribution. Jeffreys’ prior distribution is a kind of Non-informative prior distribution. This prior is used when the information about parameter not available. Non-informative Jeffreys’ prior distribution is combined with the sample information resulting the posterior distribution. Posterior distribution is used to estimate the parameter. The purposes of this research is to estimate the parameters of multivariate regression model using Bayesian method with Non-informative Jeffreys’ prior distribution. Based on the results and discussion, parameter estimation of β and Σ which were obtained from expected value of random variable of marginal posterior distribution function. The marginal posterior distributions for β and Σ are multivariate normal and inverse Wishart. However, in calculation of the expected value involving integral of a function which difficult to determine the value. Therefore, approach is needed by generating of random samples according to the posterior distribution characteristics of each parameter using Markov chain Monte Carlo (MCMC) Gibbs sampling algorithm.

Bayesian accelerated acceptance sampling plans for a lognormal lifetime distribution under Type-I censoring

Developing an accelerated acceptance sampling plan (AASP) is essential to product reliability demonstration, particularly for a product designed to have high reliability and long lifetime. Because of acceleration, a decision on lot acceptance for the product can be quickly made while significantly reducing the total cost of testing. Traditionally, the parameters of an acceptance probability function used in AASP are assumed to be known in advance. However, it is often difficult, if not impossible, to determine the exact values of these parameters. On the other hand, prior information, including earlier test results and expert opinions, may be available for potential uses in planning such tests. In this paper, Bayesian designs of AASP are considered by taking full advantage of prior information to reduce the uncertainty in these important parameters so that significant testing resources (e.g., sample size and testing time) can be saved. In particular, a product with a lognormal lifetime distribution is studied, and two types of Bayesian AASPs under Type-I censoring are developed based on the product's operating characteristic (OC) curve. The first Bayesian AASP considers the risks of the producer and the consumer simultaneously, while the second one considers only the consumer's risk. The optimal sampling plans aim at minimizing the total costs of testing. The proposed AASPs are illustrated using two numerical examples. Sensitivity analysis on the selection of prior distribution is also performed to validate the robustness of the proposed Bayesian AASPs. In addition, a comparison study shows that the proposed test plans outperform the traditional AASPs with respect to the use of testing resources.

Fitting Residual Error Structures for Growth Models in SAS PROC MCMC

In behavioral sciences broadly, estimating growth models with Bayesian methods is becoming increasingly common, especially to combat small samples common with longitudinal data. Although Mplus is becoming an increasingly common program for applied research employing Bayesian methods, the limited selection of prior distributions for the elements of covariance structures makes more general software more advantages under certain conditions. However, as a disadvantage of general software's software flexibility, few preprogrammed commands exist for specifying covariance structures. For instance, PROC MIXED has a few dozen such preprogrammed options, but when researchers divert to a Bayesian framework, software offer no such guidance and requires researchers to manually program these different structures, which is no small task. As such the literature has noted that empirical papers tend to simplify their covariance matrices to circumvent this difficulty, which is not desirable because such a simplification will likely lead to biased estimates of variance components and standard errors. To facilitate wider implementation of Bayesian growth models that properly model covariance structures, this article overviews how to generally program a growth model in SAS PROC MCMC and then demonstrates how to program common residual error structures. Full annotated SAS code and an applied example are provided.

Characterizing Dirichlet Priors

The selection of prior distributions is a problem that has been heavily discussed since Bayes and Price published their article in 1763. Conjugate priors became popular, largely because of their mathematical convenience. In this study, we justify the use of the conjugate combination of a Dirichlet prior and a multinomial likelihood by imposing a fundamental principle that we call partition invariance, alongside other requirements that are well known in the literature.[Received January 2014. Revised July 2015.]

Image denoising via Bayesian estimation of local variance with Maxwell density prior

The need for efficient image denoising methods has grown with the massive production of digital images and movies of all kinds. The distortion of images by additive white Gaussian noise (AWGN) is common during its processing and transmission. This paper is concerned with dual-tree complex wavelet-based image denoising using Bayesian techniques. Indeed, one of the cruxes of the Bayesian image denoising algorithms is to estimate the local variance of the image. Here, we employ maximum a posteriori (MAP) estimation to calculate local observed variance with Maxwell density prior for local observed variance and Gaussian distribution for noisy wavelet coefficients. Evidently, our selection of prior distribution is motivated by analytical and computational tractability. The experimental results show that the proposed method yields good denoising results.

Challenging Issues in Stochastic Calibration based on Bayesian paradigm for Building Energy Model

Bayesian calibration has been used to transform the prior distributions of unknown inputs inherited in a building energy simulation model into the trustworthy posterior distributions. It obtains the posterior distributions using a joint distribution composed of a likelihood function and prior distributions of unknown inputs in the Bayesian paradigm. In other words, it provides higher benefits in terms of a stochastic approach than the deterministic calibration and the feasibility of their calibrated results was sufficiently discussed. However, challenging issues in Bayesian calibration still remains as follows: (1) inappropriate selection of prior distributions, (2) truncated sample dataset of the likelihood functions. The aforementioned issues can increase the risks of Bayesian calibration. This paper aims to inform the risks of Bayesian calibration associated with the aforementioned issues through a reference case study. For this study, the Gaussian Process (GP) emulator, which can be regarded as a meta-model of Building Performance Simulation (BPS) tools, was used to reduce the simulation run-time. Bayesian calibration using the GP emulator was implemented with what-if scenarios considering the aforementioned issues. And then the validated models were used for a stochastic retrofit analysis of glazing systems. With the results of the estimated posterior distributions, validation, and stochastic retrofit, this paper presents Bayesian calibration issues regarding the selection of prior distributions and sample dataset of the likelihood functions.

Image Denoising via Bayesian Estimation of Statistical Parameter Using Generalized Gamma Density Prior in Gaussian Noise Model

The application of image processing in industry has shown remarkable success over the last decade, for example, in security and telecommunication systems. The denoising of natural image corrupted by Gaussian noise is a classical problem in image processing. So, image denoising is an indispensable step during image processing. This paper is concerned with dual-tree complex wavelet-based image denoising using Bayesian techniques. One of the cruxes of the Bayesian image denoising algorithms is to estimate the statistical parameter of the image. Here, we employ maximum a posteriori (MAP) estimation to calculate local observed variance with generalized Gamma density prior for local observed variance and Laplacian or Gaussian distribution for noisy wavelet coefficients. Evidently, our selection of prior distribution is motivated by efficient and flexible properties of generalized Gamma density. The experimental results show that the proposed method yields good denoising results.

A Bayesian Analysis for the Parameters of the Exponential-Logarithmic Distribution

ABSTRACT The exponential-logarithmic is a new lifetime distribution with decreasing failure rate and interesting applications in the biological and engineering sciences. Thus, a Bayesian analysis of the parameters would be desirable. Bayesian estimation requires the selection of prior distributions for all parameters of the model. In this case, researchers usually seek to choose a prior that has little information on the parameters, allowing the data to be very informative relative to the prior information. Assuming some noninformative prior distributions, we present a Bayesian analysis using Markov Chain Monte Carlo (MCMC) methods. Jeffreys prior is derived for the parameters of exponential-logarithmic distribution and compared with other common priors such as beta, gamma, and uniform distributions. In this article, we show through a simulation study that the maximum likelihood estimate may not exist except under restrictive conditions. In addition, the posterior density is sometimes bimodal when an improper prior density is used.

Comparing Bayesian and Frequentist Approaches for Multiple Outcome Mixed Treatment Comparisons

Bayesian statistical methods are increasingly popular as a tool for meta-analysis of clinical trial data involving both direct and indirect treatment comparisons. However, appropriate selection of prior distributions for unknown model parameters and checking of consistency assumptions required for modeling remain particularly challenging. We compared Bayesian and traditional frequentist statistical methods for mixed treatment comparisons with multiple binary outcomes. We searched major electronic bibliographic databases, Food and Drug Administration reviews, trial registries, and research grant databases up to December 2011 to find randomized studies published in English that examined drugs for female urgency urinary incontinence (UI) on continence, improvement in UI, and treatment discontinuation due to harm. We describe and fit fixed and random effects models in both Bayesian and frequentist statistical frameworks. In a hierarchical model of 8 treatments, we separately analyze 1 safety and 2 efficacy outcomes. We produce Bayesian and frequentist treatment ranks and odds ratios across all drug v placebo comparisons, as well as Bayesian probabilities that each drug is best overall through a weighted scoring rule that trades off efficacy and safety. In our study, Bayesian and frequentist random effects models generally suggest the same drugs as most attractive, although neither suggests any significant differences between drugs. However, the Bayesian methods more consistently identify one drug (propiverine) as best overall, produce interval estimates that are generally better at capturing all sources of uncertainty in the data, and also permit attractive "rankograms" that visually capture the probability that each drug assumes each possible rank. Bayesian methods are more flexible and their results more clinically interpretable, but they require more careful development and specialized software.

Mixtures of Gaussian wells: Theory, computation, and application

A primary challenge in unsupervised clustering using mixture models is the selection of a family of basis distributions flexible enough to succinctly represent the distributions of the target subpopulations. In this paper we introduce a new family of Gaussian well distributions (GWDs) for clustering applications where the target subpopulations are characterized by hollow (hyper-)elliptical structures. We develop the primary theory pertaining to the GWD, including mixtures of GWDs, selection of prior distributions, and computationally efficient inference strategies using Markov chain Monte Carlo. We demonstrate the utility of our approach, as compared to standard Gaussian mixture methods on a synthetic dataset, and exemplify its applicability on an example from immunofluorescence imaging, emphasizing the improved interpretability and parsimony of the GWD-based model.

Accounting for model uncertainty in estimating the relative risk of mortality associated with heat waves.

Background and Aims: Estimating the risks heat waves pose to human health is a critical part of assessing the future impact of climate change. While uncertainty in climate model predictions has been studied thoroughly, model uncertainty in estimating the subsequent health effects has not been well explored. This work assesses the contribution of model uncertainty and develops an approach to systematically incorporate model uncertainty into heat wave risk estimation. Methods: We propose a flexible class of time series models to estimate the relative risk of mortality associated with heat waves and conduct Bayesian model averaging (BMA) to account for the multiplicity of potential models. Methods are applied to data from 105 U.S. cities for the period 1987-2005. We identify those cities having a high posterior probability of increased mortality risk during heat waves, examine the heterogeneity of the posterior distributions of mortality risk across cities, assess sensitivity of the results to the selection of prior distributions, and compare our BMA results to a model selection approach. Results: Our results show that no single model best predicts risk across the majority of cities, and that for some cities heat wave risk estimation is sensitive to model choice. Conclusions: While model averaging leads to posterior distributions with increased variance as compared to statistical inference conditional on a model obtained through model selection, we find that the posterior mean of heat wave mortality risk is robust to accounting for model uncertainty over a broad class of models.

A Bayesian model averaging approach for estimating the relative risk of mortality associated with heat waves in 105 U.S. cities.

Estimating the risks heat waves pose to human health is a critical part of assessing the future impact of climate change. In this article, we propose a flexible class of time series models to estimate the relative risk of mortality associated with heat waves and conduct Bayesian model averaging (BMA) to account for the multiplicity of potential models. Applying these methods to data from 105 U.S. cities for the period 1987-2005, we identify those cities having a high posterior probability of increased mortality risk during heat waves, examine the heterogeneity of the posterior distributions of mortality risk across cities, assess sensitivity of the results to the selection of prior distributions, and compare our BMA results to a model selection approach. Our results show that no single model best predicts risk across the majority of cities, and that for some cities heat-wave risk estimation is sensitive to model choice. Although model averaging leads to posterior distributions with increased variance as compared to statistical inference conditional on a model obtained through model selection, we find that the posterior mean of heat wave mortality risk is robust to accounting for model uncertainty over a broad class of models.

The Rule for Selection of Prior Distributions for Binomial Parameter Inference Based on Hierarchical Bayes Modeling

The Rule for Selection of Prior Distributions for Binomial Parameter Inference Based on Hierarchical Bayes Modeling

Selection of prior distributions and multiscale analysis in Bayesian temperature reconstructions based on fossil assemblages

It is possible to reconstruct the past variation of an environmental variable from measured historical indicators when the modern values of the variable and the indicators are known. In a Bayesian statistical approach, the selection of a prior probability distribution for the past values of the environmental variable can then be crucial and the selection therefore should be made carefully. This is particularly the case when the data are noisy and the statistical model used is complex since the influence of the prior on the results can then be especially strong. It can be difficult to elicit the prior probability distribution from the available information, since usually there are no measured data on the past values of the variable one wants to reconstruct and different reconstructions are typically consistent with each other only at a coarse level. To overcome these difficulties we propose to use a non-informative smoothing prior, possibly in combination with an informative prior, that simply penalizes for roughness of the reconstruction as measured by the variability of its values. We believe that it can sometimes be easier to set an overall prior distribution on the roughness than to agree on a prior for the actual values of the reconstructed variable. Note that by using a smoothing prior one incorporates into the model itself the smoothing step usually done before or after the actual numerical reconstruction. Another idea proposed in this paper is to integrate the reconstruction model with a multiscale feature analysis technique known as SiZer. Multiscale analysis of the posterior distribution of the reconstructed variable makes it possible to infer its statistically significant features such as trends, maxima and minima at several different time scales. While only temperature is considered in this paper, the technique can be applied to other environmental variables.

赤池ベイズ情報量規準による事前分布の選択を考慮した逆解析に関する基礎的研究

赤池ベイズ情報量規準による事前分布の選択を考慮した逆解析に関する基礎的研究