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Establishing an accurate diagnosis is crucial in everyday clinical practice. It forms the starting point for clinical decision-making, for instance regarding treatment options or further testing. In this context, clinicians have to deal with probabilities (instead of certainties) that are often hard to quantify. During the diagnostic process, clinicians move from the probability of disease before testing (prior or pretest probability) to the probability of disease after testing (posterior or posttest probability) based on the results of one or more diagnostic tests. This reasoning in probabilities is reflected by a statistical theorem that has an important application in diagnosis: Bayes' rule. A basic understanding of the use of Bayes' rule in diagnosis is pivotal for clinicians. This rule shows how both the prior probability (also called prevalence) and the measurement properties of diagnostic tests (sensitivity and specificity) are crucial determinants of the posterior probability of disease (predictive value), on the basis of which clinical decisions are made. This article provides a simple explanation of the interpretation and use of Bayes’ rule in diagnosis.

This book provides a solution to the ecological inference problem, which has plagued users of statistical methods for over seventy-five years: How can researchers reliably infer individual-level behavior from aggregate (ecological) data? In political science, this question arises when individual-level surveys are unavailable (for instance, local or comparative electoral politics), unreliable (racial politics), insufficient (political geography), or infeasible (political history). This ecological inference problem also confronts researchers in numerous areas of major significance in public policy, and other academic disciplines, ranging from epidemiology and marketing to sociology and quantitative history. Although many have attempted to make such cross-level inferences, scholars agree that all existing methods yield very inaccurate conclusions about the world. In this volume, Gary King lays out a unique--and reliable--solution to this venerable problem. King begins with a qualitative overview, readable even by those without a statistical background. He then unifies the apparently diverse findings in the methodological literature, so that only one aggregation problem remains to be solved. He then presents his solution, as well as empirical evaluations of the solution that include over 16,000 comparisons of his estimates from real aggregate data to the known individual-level answer. The method works in practice. King's solution to the ecological inference problem will enable empirical researchers to investigate substantive questions that have heretofore proved unanswerable, and move forward fields of inquiry in which progress has been stifled by this problem.

This study presents a novel algorithm for image classification based on a quasi-Bayesian approach and the extraction of probability density functions (PDFs). First, representative PDFs are extracted from each image using its features. Next, a measure is developed to evaluate the similarity between the extracted PDFs. Finally, an algorithm is established for determining prior probabilities using fuzzy clustering techniques. By combining these improvements, we develop a more efficient algorithm for classifying image data. An image is assigned to a specific group if it has the highest value of prior probability and a similar level to that group. We explain the proposed algorithm step-by-step with a numerical example and clearly demonstrate its convergence. When applied to multiple image datasets, the proposed algorithm has shown stability and efficiency, outperforming many other statistical and machine learning methods. Additionally, we have developed a Matlab procedure to apply the proposed algorithm to real image datasets. These applications demonstrate the potential of research in various fields related to the digital revolution and artificial intelligence.

Quantification is the supervised learning task that consists of training predictors of the class prevalence values of sets of unlabelled data, and is of special interest when the labelled data on which the predictor has been trained and the unlabelled data are not IID, i.e., suffer from dataset shift. To date, quantification methods have mostly been tested only on a special case of dataset shift, i.e., prior probability shift; the relationship between quantification and other types of dataset shift remains, by and large, unexplored. In this work we carry out an experimental analysis of how current quantification algorithms behave under different types of dataset shift, in order to identify limitations of current approaches and hopefully pave the way for the development of more broadly applicable methods. We do this by proposing a fine-grained taxonomy of types of dataset shift, by establishing protocols for the generation of datasets affected by these types of shift, and by testing existing quantification methods on the datasets thus generated. One finding that results from this investigation is that many existing quantification methods that had been found robust to prior probability shift are not necessarily robust to other types of dataset shift. A second finding is that no existing quantification method seems to be robust enough to dealing with all the types of dataset shift we simulate in our experiments. The code needed to reproduce all our experiments is publicly available at https://github.com/pglez82/quant_datasetshift.

Diagnosing type 1 diabetes in adults is difficult since type 2 diabetes is the predominant diabetes type, particularly with an older age of onset (approximately >30 years). Misclassification of type 1 diabetes in adults is therefore common and will impact both individual patient management and the reported features of clinically classified cohorts. In this article, we discuss the challenges associated with correctly identifying adult-onset type 1 diabetes and the implications of these challenges for clinical practice and research. We discuss how many of the reported differences in the characteristics of autoimmune/type 1 diabetes with increasing age of diagnosis are likely explained by the inadvertent study of mixed populations with and without autoimmune aetiology diabetes. We show that when type 1 diabetes is defined by high-specificity methods, clinical presentation, islet-autoantibody positivity, genetic predisposition and progression of C-peptide loss remain broadly similar and severe at all ages and are unaffected by onset age within adults. Recent clinical guidance recommends routine islet-autoantibody testing when type 1 diabetes is clinically suspected or in the context of rapid progression to insulin therapy after a diagnosis of type 2 diabetes. In this moderate or high prior-probability setting, a positive islet-autoantibody test will usually confirm autoimmune aetiology (type 1 diabetes). We argue that islet-autoantibody testing of those with apparent type 2 diabetes should not be routinely undertaken as, in this low prior-prevalence setting, the positive predictive value of a single-positive islet antibody for autoimmune aetiology diabetes will be modest. When studying diabetes, extremely high-specificity approaches are needed to identify autoimmune diabetes in adults, with the optimal approach depending on the research question. We believe that until these recommendations are widely adopted by researchers, the true phenotype of late-onset type 1 diabetes will remain largely misunderstood. Graphical Abstract

This paper proposes a new approach for dealing with imbalanced classes and prior probability shifts in supervised classification tasks. Coupled with any feature space partitioning method, our criterion aims to compute an almost-Bayesian randomized equalizer classifier for which the maxima of the class-conditional risks are minimized. Our approach belongs to the historically well-studied field of randomized minimax criteria. Our new criterion can be considered as a self-sufficient classifier, or can be easily coupled with any pretrained Convolutional Neural Networks and Decision Trees to address the issues of imbalanced classes and prior probability shifts. Numerical experiments compare our criterion to several state-of-the-art algorithms and show the relevance of our approach when it is necessary to well classify the minority classes and to equalize the risks per class. Experiments on the CIFAR-100 database show that our criterion scales well when the number of classes is large.

Model selection is of fundamental importance to high dimensional modelling featured in many contemporary applications. Classical principles of model selection include the Bayesian principle and the Kullback–Leibler divergence principle, which lead to the Bayesian information criterion and Akaike information criterion respectively, when models are correctly specified. Yet model misspecification is unavoidable in practice. We derive novel asymptotic expansions of the two well‐known principles in misspecified generalized linear models, which give the generalized Bayesian information criterion and generalized Akaike information criterion. A specific form of prior probabilities motivated by the Kullback–Leibler divergence principle leads to the generalized Bayesian information criterion with prior probability, GBICp, which can be naturally decomposed as the sum of the negative maximum quasi‐log‐likelihood, a penalty on model dimensionality, and a penalty on model misspecification directly. Numerical studies demonstrate the advantage of the new methods for model selection in both correctly specified and misspecified models.

Most researches on regional flood frequency analysis (RFFA) have proved that the incorporation of hydrologic information (e.g., catchment attributes and flood records) from different sites in a region can provide more accurate flood estimation than using only the observed flood series at the site of concern. One kind of RFFA is based on the Bayesian method with prior information inferred from regional regression by using the generalized least squares (GLS) model, which is more flexible than other RFFA methods. However, the GLS model for regional regression is a stationary method and not suitable for coping with nonstationary prior information. In this study, in nonstationary condition, the Bayesian RFFA with the prior information inferred from regional regression by using the linear mixed effect (LME) model (i.e. a model that adds random effects to the GLS model) is investigated. Both the GLS-based and LME-based Bayesian RFFA methods have been applied to four hydrological stations within the Dongting Lake basin for comparison, and the results show that the performance of nonstationary LME-based Bayesian RFFA method is better than that of stationary GLS-based Bayesian RFFA method according to the deviance information criterion (DIC). Compared with the stationary GLS-based Bayesian RFFA method, changes in uncertainty of regression coefficients estimation of at-site flood distribution parameters are different from site to site by using the nonstationary LME-based Bayesian RFFA method. The use of nonstationary LME-based Bayesian RFFA method reduces design flood uncertainty, especially for the very small exceedance probability at the tail. This study extends the application of the Bayesian RFFA method to the nonstationary condition, which is helpful for nonstationary flood frequency analysis of ungauged sites.

Landslide susceptibility mapping is typically based on binary prediction probabilities. However, non-landslide samples in modeling datasets are often unlabeled data, and the phenomenon of class-priori shift, that is, the proportion of landslide samples frequently deviates from real-world scenarios and is spatially heterogeneous. By comparing the classification performance and predicted probability distributions across multiple unbalanced datasets with known and unknown sample proportions, this study assesses the landslide susceptibility model’s generalization ability in the context of class-prior shifts. The study investigates the potential of Bagging PU Learning, a semi-supervised learning approach, in improving the generalization performance of landslide susceptibility models and proposes the Bagging PU-GDBT algorithm. Our findings highlight the effectiveness of Bagging PU Learning in enhancing the recall of landslides and the generalization capabilities of models on unbalanced datasets. This method reduces prediction uncertainties, especially in high and very high susceptibility zones. Furthermore, results emphasize the superiority of models trained on balanced datasets with 1:1 sample ratio for landslide susceptibility mapping over those trained on unbalanced datasets.

Background In medical diagnostics, estimating post-test or posterior probabilities for disease, positive and negative predictive values, and their associated uncertainty is essential for patient care. Objective The aim of this work is to introduce a software tool developed in the Wolfram Language for the parametric estimation, visualization, and comparison of Bayesian diagnostic measures and their uncertainty. Methods This tool employs Bayes' theorem to estimate positive and negative predictive values and posterior probabilities for the presence and absence of a disease. It estimates their standard sampling, measurement, and combined uncertainty, as well as their confidence intervals, applying uncertainty propagation methods based on first-order Taylor series approximations. It employs normal, lognormal, and gamma distributions. Results The software generates plots and tables of the estimates to support clinical decision-making. An illustrative case study using fasting plasma glucose data from the National Health and Nutrition Examination Survey (NHANES) demonstrates its application in diagnosing diabetes mellitus. The results highlight the significant impact of measurement uncertainty on Bayesian diagnostic measures, particularly on positive predictive value and posterior probabilities. Conclusion The software tool enhances the estimation and facilitates the comparison of Bayesian diagnostic measures, which are critical for medical practice. It provides a framework for their uncertainty quantification and assists in understanding and applying Bayes' theorem in medical diagnostics.