Informative Prior Distributions Research Articles

Kinetic profile inference with outlier detection using support vector machine regression and Gaussian process regression

Abstract We propose an outlier-resilient Gaussian process regression (GPR) model supported by support vector machine regression (SVMR) for kinetic profile inference. GPR, being a non-parametric regression using Bayesian statistics, has advantages in that it imposes no constraints on profile shapes and can be readily used to integrate different kinds of diagnostics, while it is vulnerable to the presence of even a single outlier among a measured dataset. As an outlier classifier, an optimized SVMR is developed based only on the measurements. Hyper-parameters of the developed GPR model with informative prior distributions are treated in two different ways, i.e. maximum a posteriori (MAP) estimator and marginalization using a Markov Chain Monte Carlo sampler. Our SVMR-supported GPR model is applied to infer ion temperature Ti profiles using measured data from the KSTAR charge exchange spectroscopy system. The GPR-inferred Ti profiles with and without an outlier are compared and show prominent improvement when the outlier is removed by the SVMR. Ti profiles inferred with the MAP estimator and the marginalization scheme are compared. They are noticeably different when observation uncertainties are not small enough, and the marginalization scheme generally provides a smoother profile.

An extensive re-evaluation of evidence and analyses of the Randomised Badger Culling Trial (RBCT) I: Within proactive culling areas.

Here, in the first of two investigations, we evaluate and extend the analyses of the Randomised Badger Culling Trial (RBCT) to estimate the effectiveness of proactive badger culling for reducing incidence of tuberculosis (TB) in cattle within culling areas. Using previously reviewed, publicly available data, alongside frequentist and Bayesian approaches, we re-estimate culling effects for confirmed incidence of herd breakdowns (TB incidents in cattle) within proactive culling areas. We appraise the varying assumptions and statistical structures of individual models to determine model appropriateness. Our re-evaluation of frequentist models provides results consistent with peer-reviewed analyses of RBCT data, due to the consistency of beneficial effects across three analysis periods. Furthermore, well-fitting Bayesian models with weakly informative prior distribution assumptions produce high probabilities (91.2%-99.5%) of beneficial effects of proactive culling on confirmed herd breakdowns within culling areas in the period from the initial culls (between 1998 and 2002) until 2005. Similarly high probabilities of beneficial effects were observed post-trial (from 1 year after last culls until March 2013). Thus, irrespective of statistical approach or study period, we estimate substantial beneficial effects of proactive culling within culling areas, consistent with separate, existing, peer-reviewed analyses of the RBCT data.

Optimized multi-point hemispherical grid model with adaptive grid division based on the prior information of multipath error

The multi-point hemispherical grid model (MHGM) utilizes residual of double-differenced observations to extract precise multipath error information. It models the entire network of multipath error effects across different stations to achieve effective error correction. However, because all the parameters are estimated collectively using the least squares method, the increased number of grid point parameters can significantly consume memory, CPU, and other computing resources required for modeling. In response to the computational resource consumption challenge associated with fixed-resolution MHGM in multi-station applications, a space domain adaptive grid division method is proposed to optimize the modeling of multipath errors. This approach utilizes prior distribution information of multipath errors to optimize the grid structure. It reduces the number of grids in areas where multipath errors exhibit minimal changes, and provides detailed parameterization for areas with significant variations. Experimental results demonstrate the effectiveness of this method in significantly reducing the number of estimated parameters using MHGM. In statistical analysis of double-differenced phase observation residuals with fixed ambiguities, as the number of estimated parameters in the MHGM decreases to only 24.6 % of the fixed-resolution approach, memory usage during parameter estimation remains a mere 6 % of that required in the fixed-resolution approach. This highlights its potential value in mitigating multipath errors when modeling GNSS large-scale network data.

Residual squeeze-and-excitation convolutional auto-encoder for fault detection and diagnosis in complex industrial processes

Recently, deep learning has attracted increasing attention in process monitoring. However, it is still a big challenge to extract effective features of process data for fault detection and diagnosis (FDD) by using deep learning-based methods. In this study, a novel deep learning-based model, residual squeeze-and-excitation convolutional auto-encoder (RSECAE) is proposed for FDD in complex industrial processes. A hybrid structure that integrates convolution calculation and auto-encoder is proposed to construct multiple convolution layers and deconvolution layers in the encoder and decoder respectively. Thus, RSECAE can learn effective features from the process signals in an unsupervised manner. At the same time, multiple residual squeeze-and-excitation networks (RSENets) are embedded in the deep network, which effectively preserves the critical feature representations of the convolution layer and transmits them to the decoder in the form of residuals. Finally, maximum mean discrepancy (MMD) term is utilized in RSECAE to reduce the distribution difference between the process signals and the learned features. This enables RSECAE to extract effective features with prior distribution information. RSECAE provides a new solution for process FDD for complicated industrial processes. Two industrial cases are adopted to validate the effectiveness of the RSECAE in process FDD. The experimental results demonstrate the effectiveness and superiority of RSECAE in process fault detection and fault diagnosis.

A Bayesian approach to data-driven multi-stage stochastic optimization

Aimed at sufficiently utilizing available data and prior distribution information, we introduce a data-driven Bayesian-type approach to solve multi-stage convex stochastic optimization, which can easily cope with the uncertainty about data process’s distributions and their inter-stage dependence. To unravel the properties of the proposed multi-stage Bayesian expectation optimization (BEO) problem, we establish the consistency of optimal value functions and solutions. Two kinds of algorithms are designed for the numerical solution of single-stage and multi-stage BEO problems, respectively. A queuing system and a multi-stage inventory problem are adopted to numerically demonstrate the advantages and practicality of the new framework and corresponding solution methods, compared with the usual formulations and solution methods for stochastic optimization problems.

Underwater image enhancement based on global features and prior distribution guided

Underwater images often suffer from substantial image blur and color distortion due to the variability of water conditions and the physical location of optical equipment, which significantly impacts the underwater intelligent system's environmental perception. Standard methods exhibit limited generalization capabilities, leading to considerable performance fluctuations when handling images with uncontrolled degradation. In this research, we leverage global features and the prior distribution of ground truth images to guide our enhancement model, introducing a novel conditional Variational Auto-Encoder-based model, named UWG-VAE, to address these challenges. UWG-VAE enhances model controllability by incorporating prior distribution information and classes of degraded styles into the decoder of the enhancement model. We assess the performance of UWG-VAE in underwater image enhancement tasks across four challenging real underwater image datasets, comparing it to state-of-the-art models. UWG-VAE demonstrates a substantial enhancement in visual quality, with notable improvements in UIQM, UCIQE, and URanker evaluation metrics when compared to existing state-of-the-art models.

Beyond the Classical Type I Error: Bayesian Metrics for Bayesian Designs Using Informative Priors

There is growing interest in Bayesian clinical trial designs with informative prior distributions, for example for extrapolation of adult data to pediatrics, or use of external controls. While the classical Type I error is commonly used to evaluate such designs, it cannot be strictly controlled and it is acknowledged that other metrics may be more appropriate. We focus on two common situations—borrowing control data or information on the treatment contrast—and discuss several fully probabilistic metrics to evaluate the risk of false positive conclusions. Each metric requires specification of a design prior, which can differ from the analysis prior and permits understanding of the behavior of a Bayesian design under scenarios where the analysis prior differs from the true data generation process. The metrics include the average Type I error and the pre-posterior probability of a false positive result. When borrowing control data, our empirical cases demonstrate that the average Type I error is asymptotically controlled (in certain cases strictly) when the analysis and design prior coincide. We illustrate use of these Bayesian metrics with real applications, and discuss how they could facilitate discussions between sponsors, regulators and other stakeholders about the appropriateness of Bayesian borrowing designs for pivotal studies.

Comparison of Bayesian and frequentist monitoring boundaries motivated by the Multiplatform Randomized Clinical Trial.

The coronavirus disease 2019 pandemic highlighted the need to conduct efficient randomized clinical trials with interim monitoring guidelines for efficacy and futility. Several randomized coronavirus disease 2019 trials, including the Multiplatform Randomized Clinical Trial (mpRCT), used Bayesian guidelines with the belief that they would lead to quicker efficacy or futility decisions than traditional "frequentist" guidelines, such as spending functions and conditional power. We explore this belief using an intuitive interpretation of Bayesian methods as translating prior opinion about the treatment effect into imaginary prior data. These imaginary observations are then combined with actual observations from the trial to make conclusions. Using this approach, we show that the Bayesian efficacy boundary used in mpRCT is actually quite similar to the frequentist Pocock boundary. The mpRCT's efficacy monitoring guideline considered stopping if, given the observed data, there was greater than 99% probability that the treatment was effective (odds ratio greater than 1). The mpRCT's futility monitoring guideline considered stopping if, given the observed data, there was greater than 95% probability that the treatment was less than 20% effective (odds ratio less than 1.2). The mpRCT used a normal prior distribution that can be thought of as supplementing the actual patients' data with imaginary patients' data. We explore the effects of varying probability thresholds and the prior-to-actual patient ratio in the mpRCT and compare the resulting Bayesian efficacy monitoring guidelines to the well-known frequentist Pocock and O'Brien-Fleming efficacy guidelines. We also contrast Bayesian futility guidelines with a more traditional 20% conditional power futility guideline. A Bayesian efficacy and futility monitoring boundary using a neutral, weakly informative prior distribution and a fixed probability threshold at all interim analyses is more aggressive than the commonly used O'Brien-Fleming efficacy boundary coupled with a 20% conditional power threshold for futility. The trade-off is that more aggressive boundaries tend to stop trials earlier, but incur a loss of power. Interestingly, the Bayesian efficacy boundary with 99% probability threshold is very similar to the classic Pocock efficacy boundary. In a pandemic where quickly weeding out ineffective treatments and identifying effective treatments is paramount, aggressive monitoring may be preferred to conservative approaches, such as the O'Brien-Fleming boundary. This can be accomplished with either Bayesian or frequentist methods.

Assessing risk factors of bypass graft surgery through the implementation of Bayesian and non-Bayesian methodologies

The aim of this research is to determine the key risk factors that influence the need for Bypass Graft surgery (BGS). A cohort of one hundred adult patients underwent coronary angiography operations at the Erbil Cardiac Center from January 2016 to December 2020. Multiple logistic regression (MLR) analysis was used to identify the determinants of BGS. Upon detailed statistical analysis, a significant association with BGS was found for several variables, including Age, BMI, Eosinophil, WBC, DBP, HbA1c, MCH, Blood Sugar, PLT, and T3. It was found that the significant variables contributing the most to predicting the likelihood of Bypass Graft surgery through Bayesian logistic regression analysis were Age, WBC, and MCH. These variables were identified as the pivotal risk factors associated with the probability of undergoing BGS. Age, DBP, and Blood Sugar had minimal influence on the likelihood of Bypass surgery. However, for each unit increase in WBC, the odds of surgery reduced by 38 %, and for MCH, the odds reduced by 19 %. DBP and Blood Sugar had little effect on Bypass surgery (Bayesian odds ratio close to 1). This study revealed that Bayesian logistic models offer superiority over classical logistic models. The Bayesian-based models provide more accurate predictions by incorporating informative prior distributions. To improve the accuracy and generalizability of the findings, future research should consider a larger sample size and incorporate informative prior distributions within the Bayesian framework. Also, machine learning algorithms can offer new insights into BGS outcomes by considering various variables and assessing model sensitivity.

Exploring transportation equity issues for persons with disabilities: The impact of gender on mobility and accessibility indicators

IntroductionPersons with disabilities still face numerous barriers in their everyday lives, especially in transportation. The problem of access to transportation is identified through fewer daily trips and more barriers in public transport. These problems indicate transportation equity issues among persons with disabilities. Another issue recognized in previous studies was a gender imbalance in the samples that could indicate gender equality problems among persons with disabilities. This paper aims to investigate mobility and accessibility indicators among persons with disabilities, with a particular emphasis on the impact of gender, in order to explore two critical transportation equity issues: inclusivity and social justice. MethodsThe mobility and accessibility indicators analyzed in this research were the Number of daily trips, Mobility satisfaction, Infrastructure accessibility satisfaction, and Modal distributions of daily trips. The data about indicators was collected by an online version of a questionnaire. The research was performed on 100 persons with disabilities from Serbia, equally distributed by gender. The most common recorded diagnosis of the respondents was spinal cord injury. ResultsIn general, respondents made 3.26 trips per day. At two transportation modes, we identified significant differences between males and females. Males frequently tripped as passenger car drivers, and females often traveled as passengers in passenger cars. Additionally, nine Bayesian linear regression models with weak informative prior distributions brought much helpful information. ConclusionsThe obtained results bring stakeholders valuable information in creating policies, practices, and recommendations to achieve specific goals related to transportation equity. Improving flexibility and time efficiency of females’ trips and resolving the problem of females’ fear of abuse and harassment in taxi cars were recognized as the two most critical topics.

Meta-Analysis of Normal Reference Values for Right and Left Ventricular Quantification by Cardiovascular Magnetic Resonance.

Cardiovascular magnetic resonance (CMR) reference values are relied upon to accurately diagnose left ventricular (LV) and right ventricular (RV) pathologies. To date, reference values have been derived from modest sample sizes with limited patient diversity and attention to 1 but not both commonly used tracing techniques for papillary muscles and trabeculations. We sought to overcome these limitations by meta-analyzing normal reference values for CMR parameters stemming from multiple countries, vendors, analysts, and patient populations. We comprehensively extracted published and unpublished data from studies reporting CMR parameters in healthy adults. A steady-state free-precession short-axis stack at 1.5T or 3T was used to trace either counting the papillary muscles and trabeculations in the LV volume or mass. We used a novel Bayesian hierarchical meta-analysis model to derive the pooled lower and upper reference values for each CMR parameter. Our model accounted for the expected differences between tracing techniques by including informative prior distributions from a large external data set. A total of 254 studies from 25 different countries were systematically reviewed, representing 12 812 healthy adults, of which 52 were meta-analyzed. For LV parameters counting papillary muscles and trabeculations in the LV volume, pooled normative reference ranges in men and women, respectively, were as follows: LV ejection fraction of 52% to 73% and 54% to 75%, LV end-diastolic volume index of 60 to 109 and 56 to 96 mL/m2, LV end-systolic volume index of 18 to 45 and 16 to 38 mL/m2, and LV mass index of 41 to 76 and 33 to 57 g/m2. For LV parameters counting papillary muscles and trabeculations in the LV mass, pooled normative reference ranges in men and women, respectively, were as follows: LV ejection fraction of 57% to 74% and 57% to 75%, LV end-diastolic volume index of 60 to 97 and 55 to 88 mL/m2, LV end-systolic volume index of 18 to 37 and 15 to 34 mL/m2, and LV mass index of 50 to 83 and 38 to 65 g/m2. For RV parameters, pooled normative reference ranges in men and women, respectively, were as follows: RV ejection fraction of 47% to 68% and 49% to 71%, RV end-diastolic volume index of 64 to 115 and 57 to 99 mL/m2, RV end-systolic volume index of 23 to 52 and 18 to 42 mL/m2, and RV mass index of 14 to 29 and 13 to 25 g/m2. Our Bayesian hierarchical meta-analysis provides normative reference values for CMR parameters of LV and RV size, systolic function, and mass, encompassing both tracing techniques across a diverse multinational sample of healthy men and women.

Inhibition of Receptor-Interacting Protein Kinase1 in Chronic Plaque Psoriasis: A Multicenter, Randomized, Double-Blind, Placebo-Controlled Study.

Receptor-interacting protein kinase1 (RIPK1), a key mediator of inflammation through necroptosis and proinflammatory cytokine production, may play a role in the pathogenesis of immune-mediated inflammatory diseases such as chronic plaque psoriasis. An experimental medicine study of RIPK1 inhibition with GSK2982772 immediate-release formulation at doses up to 60mg three times daily in mild to moderate plaque psoriasis indicated that efficacy may be improved with higher trough concentrations of GSK2982772. This multicenter, randomized, double-blind, placebo-controlled, repeat-dose study (NCT04316585) assessed the efficacy, safety, pharmacokinetics, and pharmacodynamics of 960mg GSK2982772 (once-daily modified-release formulation) in patients with moderate to severe plaque psoriasis. Twenty-nine patients were randomized 2:1 to GSK2982772 (N = 19) or placebo (N = 10) for 12weeks. GSK2982772 was well tolerated with trough concentrations greater than tenfold higher than the previous phase1 study with immediate release. Despite near complete RIPK1 target engagement in blood and modest reduction in circulating inflammatory cytokines, the proportion of patients achieving 75% improvement from baseline in Psoriasis Area Severity Index score at week12 was similar between GSK2982772 and placebo (posterior median 1.8% vs 4.9%, respectively), with an estimated median treatment difference of - 2.3%. This analysis incorporated historical placebo data through the use of an informative prior distribution on the placebo arm. Week4 changes in skin biopsy gene expression suggested sufficient local drug exposure to elicit a pharmacodynamic response. Administration of the RIPK1 inhibitor GSK2982772 to patients with moderate to severe plaque psoriasis did not translate into meaningful clinical improvements.

Complemented subspace-based weighted collaborative representation model for imbalanced learning

Collaborative representation-based classifiers (CRCs) have demonstrated remarkable classification performance in various pattern recognition fields. However, their success heavily relies on a balanced class distribution. More radically, significantly skewed class distributions may severely affect the effectiveness of the CRCs. To address this limitation, in this study, a complemented subspace-based regularization technique is integrated into the CRC framework for tackling imbalanced classification tasks. To enhance recognition accuracy for minority classes, a class weight learning algorithm is presented, in which the weight of each class is adaptively obtained according to the prior distribution information of the original data. Furthermore, the proposed model provides an efficient closed-form solution, ensuring computational efficiency comparable to CRCs. Extensive experiments on diverse imbalanced datasets are carried out to substantiate the superiority of our method over numerous advanced imbalanced learning algorithms.

MetaNorm: incorporating meta-analytic priors into normalization of NanoString nCounter data.

Non-informative or diffuse prior distributions are widely employed in Bayesian data analysis to maintain objectivity. However, when meaningful prior information exists and can be identified, using an informative prior distribution to accurately reflect current knowledge may lead to superior outcomes and great efficiency. We propose MetaNorm, a Bayesian algorithm for normalizing NanoString nCounter gene expression data. MetaNorm is based on RCRnorm, a powerful method designed under an integrated series of hierarchical models that allow various sources of error to be explained by different types of probes in the nCounter system. However, a lack of accurate prior information, weak computational efficiency, and instability of estimates that sometimes occur weakens the approach despite its impressive performance. MetaNorm employs priors carefully constructed from a rigorous meta-analysis to leverage information from large public data. Combined with additional algorithmic enhancements, MetaNorm improves RCRnorm by yielding more stable estimation of normalized values, better convergence diagnostics and superior computational efficiency. R Code for replicating the meta-analysis and the normalization function can be found at github.com/jbarth216/MetaNorm.

Bayesian random-effects meta-analysis with empirical heterogeneity priors for application in health technology assessment with very few studies.

In Bayesian random-effects meta-analysis, the use of weakly informative prior distributions is of particular benefit in cases where only a few studies are included, a situation often encountered in health technology assessment (HTA). Suggestions for empirical prior distributions are available in the literature but it is unknown whether these are adequate in the context of HTA. Therefore, a database of all relevant meta-analyses conducted by the Institute for Quality and Efficiency in Health Care (IQWiG, Germany) was constructed to derive empirical prior distributions for the heterogeneity parameter suitable for HTA. Previously, an extension to the normal-normal hierarchical model had been suggested for this purpose. For different effect measures, this extended model was applied on the database to conservatively derive a prior distribution for the heterogeneity parameter. Comparison of a Bayesian approach using the derived priors with IQWiG's current standard approach for evidence synthesis shows favorable properties. Therefore, these prior distributions are recommended for future meta-analyses in HTA settings and could be embedded into the IQWiG evidence synthesis approach in the case of very few studies.

Robust incorporation of historical information with known type I error rate inflation.

Bayesian clinical trials can benefit from available historical information through the specification of informative prior distributions. Concerns are however often raised about the potential for prior-data conflict and the impact of Bayes test decisions on frequentist operating characteristics, with particular attention being assigned to inflation of type I error (TIE) rates. This motivates the development of principled borrowing mechanisms, that strike a balance between frequentist and Bayesian decisions. Ideally, the trust assigned to historical information defines the degree of robustness to prior-data conflict one is willing to sacrifice. However, such relationship is often not directly available when explicitly considering inflation of TIE rates. We build on available literature relating frequentist and Bayesian test decisions, and investigate a rationale for inflation of TIE rate which explicitly and linearly relates the amount of borrowing and the amount of TIE rate inflation in one-arm studies. A novel dynamic borrowing mechanism tailored to hypothesis testing is additionally proposed. We show that, while dynamic borrowing prevents the possibility to obtain a simple closed-form TIE rate computation, an explicit upper bound can still be enforced. Connections with the robust mixture prior approach, particularly in relation to the choice of the mixture weight and robust component, are made. Simulations are performed to show the properties of the approach for normal and binomial outcomes, and an exemplary application is demonstrated in a casestudy.

Time-dependent reliability assessment of existing concrete bridges with varying knowledge levels by proof load testing

In the evaluation of existing bridges and viaducts, relying solely on a desk study is often inadequate for determining their structural reliability. Performing a proof load test provides valuable field data that offers detailed information about the structural integrity. However, the relation between the magnitude of the load and the structural reliability is not immediately clear. This study addresses the challenges associated with determining the target load and highlights the uncertainties that play a key role. A case study is presented that shows the time-dependent character of the structural reliability and the influence of an informative and a weakly informative prior distribution in a Bayesian context. It is shown how both past traffic loads and a proof load test may contribute to the proven strength of a structure. The described method provides a starting point towards a flexible approach for proof load testing in which structure-specific knowledge levels and requirements are considered.

Discriminative multimodal learning via conditional priors in generative models

Deep generative models with latent variables have been used lately to learn joint representations and generative processes from multi-modal data, which depict an object from different viewpoints. These two learning mechanisms can, however, conflict with each other and representations can fail to embed information on the data modalities. This research studies the realistic scenario in which all modalities and class labels are available for model training, e.g. images or handwriting, but where some modalities and labels required for downstream tasks are missing, e.g. text or annotations. We show, in this scenario, that the variational lower bound limits mutual information between joint representations and missing modalities. We, to counteract these problems, introduce a novel conditional multi-modal discriminative model that uses an informative prior distribution and optimizes a likelihood-free objective function that maximizes mutual information between joint representations and missing modalities. Extensive experimentation demonstrates the benefits of our proposed model, empirical results show that our model achieves state-of-the-art results in representative problems such as downstream classification, acoustic inversion, and image and annotation generation.

Incorporation of healthy volunteers data on receptor occupancy into a phase II proof-of-concept trial using a Bayesian dynamic borrowing design.

Receptor occupancy in targeted tissues measures the proportion of receptors occupied by a drug at equilibrium and is sometimes used as a surrogate of drug efficacy to inform dose selection in clinical trials. We propose to incorporate data on receptor occupancy from a phase I study in healthy volunteers into a phase II proof-of-concept study in patients, with the objective of using all the available evidence to make informed decisions. A minimal physiologically based pharmacokinetic modeling is used to model receptor occupancy in healthy volunteers and to predict it in the patients of a phase II proof-of-concept study, taking into account the variability of the population parameters and the specific differences arising from the pathological condition compared to healthy volunteers. Then, given an estimated relationship between receptor occupancy and the clinical endpoint, an informative prior distribution is derived for the clinical endpoint in both the treatment and control arms of the phase II study. These distributions are incorporated into a Bayesian dynamic borrowing design to supplement concurrent phase II trial data. A simulation study in immuno-inflammation demonstrates that the proposed design increases the power of the study while maintaining a type I error at acceptable levels for realistic values of the clinicalendpoint.

Reducing Uncertainty in Site-specific Stress Prediction for CO2 Storage in the North Sea, Norway

Two CO2 storage sites located in the western Norwegian North Sea (NNS), called Aurora and Smeaheia, are currently under construction and assessment respectively. In geological storage of CO2, the in situ minimum horizontal stress is an essential input parameter for assessment of both containment and induced seismic risks [1]. To infer the stress states at certain depths at a site where no data is available, the standard approach is to perform a classical linear regression on stress data versus depth and treat the fitted trend line as the best site-specific stress predictions along depth [2]. However, stress data are often highly limited at CO2 storage sites; for example, Aurora and Smeaheia have only five in situ stress measurements available at best respectively. Such limited data may severely underrepresent the true stress distribution at one site. Data scarcity coupled with measurement error and spatial variability poses a challenge to reliable stress prediction, and hence it is crucial to quantify and reduce uncertainty in site-specific stress prediction for CO2 storage. Stress uncertainty is actually the required input information in the more rational probabilistic risk assessment framework. A natural solution to reducing uncertainty is to integrate stress information from other sources.
 On the Norwegian continental shelf, extensive data has been accumulated from previous petroleum projects. Of the publicly available NPD stress database, Figure 1a shows the distribution of versus depth (< 3,000 m) for each site within the study area containing Aurora and Smeaheia, and reveals a certain degree of similarity between the stress trends at the 11 sites. Such similarity, aligned with other published results [2], may be attributed to the relaxed sedimentary basins where gravitational loading dominates the lateral stress distribution rather than tectonic components, with the between-site variation arising from differences in the geological conditions and pore pressures [3]. When facing limited data for a site like Aurora and Smeaheia, the current approach is often to either directly use the stress trend from other sites having richer data or expand the coverage area to include more data. Such semi-subjective information borrowing approach, although effective in many cases, may lead to overly confident stress predictions as it fails to account for possible between-site heterogeneity in stress trends.
 Bayesian inference has been widely used as a rigorous and powerful statistical approach for quantifying uncertainty, as well as combining information from different sources via informative prior distributions. Hence, historical stress data may be integrated into Bayesian analysis of site-specific data in the form of prior distributions, with stress uncertainties being quantified and updated as the posterior distributions [4, 5]. When developing prior distributions for site-specific stress prediction, it may be tempting to combine all historical stress data for a holistic Bayesian analysis, yet such complete pooling approach may give an overconfident summary of prior information in that it ignores the possible stress heterogeneity between sites.
 This paper presents a Bayesian hierarchical (i.e., partial pooling) model (BHM) that explicitly accounts for between-site heterogeneity/similarity when constructing prior distributions from historical stress data, and demonstrates how the proposed model effectively borrows historical information to reduce uncertainty in site-specific stress prediction for CO2 storage in the NNS study area. Figures 1b illustrates the prior predictions of versus depth at the Aurora site from the Bayesian complete and partial pooling models. Although the complete pooling model gives less uncertain stress predictions than the partial pooling model as indicated by the narrower 90% prediction intervals (PIs), it does not well capture the five unseen stress measurements at Aurora in that two out of five stress values fall outside the 90% PIs. This suggests that complete pooling analysis indeed gives overconfident prior distributions out of the NPD database, and is thus not suitable for integrating historical data into site-specific stress prediction for CO2 storage in the NNS. On the other hand, the partial pooling model gives fairly good prior predictions of the five unseen stress values at Aurora, albeit with larger uncertainties. This result demonstrates the effectiveness of BHM as a framework for formulating proper informative priors from historical data, and an encouraging implication is that probabilistic risk assessment is allowed even with no site-specific stress data at this storage site, which is not possible if external information is not integrated properly. Figure 1c shows the posterior stress predictions updated with the five site-specific stress values from the two Bayesian models in question. After incorporating the site-specific data, the complete pooling model still over-predicts the two stress values at depths with barely noticeable updating, while partial pooling gives considerably more accurate stress predictions with reduced uncertainty.