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Belief, Evidence, and Uncertainty: Problems of Epistemic Inference (SpringerBriefs in Philosophy #0)
by Prasanta S. Bandyopadhyay Gordon Brittan Jr. Mark L. TaperThis work breaks new ground by carefully distinguishing the concepts of belief, confirmation, and evidence and then integrating them into a better understanding of personal and scientific epistemologies. It outlines a probabilistic framework in which subjective features of personal knowledge and objective features of public knowledge have their true place. It also discusses the bearings of some statistical theorems on both formal and traditional epistemologies while showing how some of the existing paradoxes in both can be resolved with the help of this framework.This book has two central aims: First, to make precise a distinction between the concepts of confirmation and evidence and to argue that failure to recognize this distinction is the source of certain otherwise intractable epistemological problems. The second goal is to demonstrate to philosophers the fundamental importance of statistical and probabilistic methods, at stake in the uncertain conditions in which for the most part we lead our lives, not simply to inferential practice in science, where they are now standard, but to epistemic inference in other contexts as well. Although the argument is rigorous, it is also accessible. No technical knowledge beyond the rudiments of probability theory, arithmetic, and algebra is presupposed, otherwise unfamiliar terms are always defined and a number of concrete examples are given. At the same time, fresh analyses are offered with a discussion of statistical and epistemic reasoning by philosophers. This book will also be of interest to scientists and statisticians looking for a larger view of their own inferential techniques.The book concludes with a technical appendix which introduces an evidential approach to multi-model inference as an alternative to Bayesian model averaging.
Belief Functions: 6th International Conference, BELIEF 2021, Shanghai, China, October 15–19, 2021, Proceedings (Lecture Notes in Computer Science #12915)
by Thierry Denœux Eric Lefèvre Zhunga Liu Frédéric PichonThis book constitutes the refereed proceedings of the 6th International Conference on Belief Functions, BELIEF 2021, held in Shanghai, China, in October 2021. The 30 full papers presented in this book were carefully selected and reviewed from 37 submissions. The papers cover a wide range on theoretical aspects on mathematical foundations, statistical inference as well as on applications in various areas including classification, clustering, data fusion, image processing, and much more.
Belief Functions: 7th International Conference, BELIEF 2022, Paris, France, October 26–28, 2022, Proceedings (Lecture Notes in Computer Science #13506)
by Sylvie Le Hégarat-Mascle Isabelle Bloch Emanuel AldeaThis book constitutes the refereed proceedings of the 7th International Conference on Belief Functions, BELIEF 2022, held in Paris, France, in October 2022.The theory of belief functions is now well established as a general framework for reasoning with uncertainty, and has well-understood connections to other frameworks such as probability, possibility, and imprecise probability theories. It has been applied in diverse areas such as machine learning, information fusion, and pattern recognition. The 29 full papers presented in this book were carefully selected and reviewed from 31 submissions. The papers cover a wide range on theoretical aspects on mathematical foundations, statistical inference as well as on applications in various areas including classification, clustering, data fusion, image processing, and much more.
Belief Revision in Non-Classical Logics (SpringerBriefs in Computer Science)
by Márcio Moretto RibeiroSince the advent of the Semantic Web, interest in the dynamics of ontologies (ontology evolution) has grown significantly. Belief revision presents a good theoretical framework for dealing with this problem; however, classical belief revision is not well suited for logics such as Description Logics.Belief Revision in Non-Classical Logics presents a framework which can be applied to a wide class of logics that include – besides most Description Logics such as the ones behind OWL – Horn Logic and Intuitionistic logic, amongst others. The author also presents algorithms for the most important constructions in belief bases. Researchers and practitioners in theoretical computing will find this an invaluable resource.
Beliefs: A Hidden Variable In Mathematics Education? (Mathematics Education Library #31)
by Gilah C. Leder Erkki Pehkonen Günter TörnerThis book focuses on aspects of mathematical beliefs, from a variety of different perspectives. Current knowledge of the field is synthesized and existing boundaries are extended. The volume is intended for researchers in the field, as well as for mathematics educators teaching the next generation of students.
Beliefs und selbstreguliertes Lernen: Eine Studie in Projektkursen der Mathematik in der gymnasialen Oberstufe (Studien zur theoretischen und empirischen Forschung in der Mathematikdidaktik)
by Hans-Jürgen StoppelHans-Jürgen Stoppel beschäftigt sich im Rahmen von Projektkursen der gymnasialen Oberstufe in Nordrhein-Westfalen mit der Entwicklung von epistemologischen Beliefs und selbstreguliertem Lernen von Schülerinnen und Schülern. Mithilfe von Mixed Methods untersucht er als Forscher und Lehrer über ein Schuljahr hinweg in Projektkursen zu Codierung und Kryptographie detailliert epistemologische Beliefs und ihre Veränderungen in Verbindung mit selbstreguliertem Lernen. Er beschreibt die Auffassung von Mathematik der Schülerinnen und Schüler als Komponente epistemologischer Beliefs im Hinblick auf die Definition von Mathematik, den Erwerb mathematischen Wissens und des mathematischen Verständnisses sowie die entsprechenden Veränderungen. Die Vertiefung des Kursthemas erlaubt es, die Ergebnisse auch zu Schlüssen auf den Übergang von der Schule zur Hochschule heranzuziehen.
Beliefs von Lehrerinnen und Lehrern der Sekundarstufen zum Visualisieren im Mathematikunterricht (Freiburger Empirische Forschung in der Mathematikdidaktik)
by Angela SchmitzAngela Schmitz analysiert die Überzeugungen und Ziele von Mathematiklehrkräften der Sekundarstufen I und II zum Visualisieren, einem wesentlichen Element des Betreibens von Mathematik. Sie untersucht die Sichtweisen der Lehrkräfte auf Visualisierung im Unterricht und inwiefern sich diese zwischen verschiedenen mathematischen Themengebieten unterscheiden. Dabei betrachtet die Autorin die Bruchrechnung, Algebra, Funktionen und Analysis und zeigt potenzielle Zusammenhänge zwischen den Beliefs von Lehrerinnen und Lehrern, verschiedenen Themengebieten und unterschiedlichen Überzeugungsfeldern auf.
Beliefs von Lehrkräften zum Lehren und Lernen von Arithmetik (Freiburger Empirische Forschung in der Mathematikdidaktik)
by Katinka BräunlingKatinka Bräunling untersucht subjektive Vorstellungen von Lehrkräften in Bezug auf den mathematischen Teilbereich der Arithmetik. Ausgehend davon, dass diese Vorstellungen (Beliefs) von Lehrerinnen und Lehrern für ihre Unterrichtsplanung und ihr Handeln im Unterricht relevant sind, stellt die Autorin die Identifikation von Beliefs und Beliefsystemen in den Mittelpunkt der empirischen Studie. In einem Mixed-Method-Design analysiert sie qualitativ und quantitativ erhobene Daten angehender und erfahrener Lehrkräfte und setzt deren Beliefsysteme in Beziehung zueinander. Dabei werden diese Systeme in ihrer Struktur, ihrer längerfristigen Entwicklung sowie in Bezug auf ihre Handlungsrelevanz betrachtet.
The Belousov-Zhabotinskii Reaction (Lecture Notes in Biomathematics #10)
by J.J. TysonIn 1958 B. P. Belousov discovered that the oxidation of citric acid by bromate in the presence of cerium ions does not proceed to equilibrium methodically and uniformly, like most chemical reactions, but rather oscillates with clocklike precision between a yellow and colorless state. See Fig. 11. 1, p. 30. A. M. Zhabotinskii followed up on Belousov's original observation and in 1964 his first investigations appeared in the Russian journal Biofizika. Though H. Degn (in Copenhagen at the time) knew of Zhabotinskii's work and published his own account of the mechanism of oscillation in Nature (1967), this interesting reaction attracted little attention among Western scientists until 1968, when Zhabotinskii and his coworkers and Busse (from Braunschweig, W. Germany) reported on their work at an international conference on biological and biochemical oscillators held in Prague. Shortly thereafter appeared· a flurry of papers on temporal oscillations and spatial patterns in this reaction system. Vavilin and Zhabotinskii (1969) [and later Kasperek and Bruice (1971)] studied the kinetics of the oxidation 3 of Ce+ by Br0 and the oxidation of organic species by Ce+4. Busse (1969) 3 reported his observation of colored bands of chemical activity propagating up and down in a long tube of unstirred solution. Zaikin and Zhabotinskii (1970) observed circular chemical waves in thin layers of solution.
“The Belt and Road” International Migration of Asia: Research on Multilateral Population Security (China Perspectives)
by MI Hong LI Yuan MA QiyiniThe book studies multilateral population security issues and relevant governance strategies caused by international migration in the countries impacted by China’s Belt and Road initiative and their border areas.Buttressed by solid data mining and policy analysis, the title looks into the demographic trends of international migration in China and some Asian Belt and Road countries and stresses the urgency for more effective governance practices. Seeking to address the population security crisis triggered by the Initiative, the authors propose the idea of “multilateral population security governance”, grounded in the real-world challenges facing Belt and Road countries while also drawing on experiences of migration governance in western countries. As a new governance model, it calls for cross-border joint action and takes into consideration pertinent factors including economy, politics, culture, religion and commerce. Several case studies and comparative studies are offered in the chapters to illuminate the significance and effect of this cooperative mechanism.The book will be of interest to researchers and government officials interested in non-traditional security, international migration and formal demography as well as topics on population, resources and environment.
“The Belt and Road” International Migration of Asia: Research on Multilateral Population Security (China Perspectives)
by MI Hong LI Yuan MA QiyiniThe book studies multilateral population security issues and relevant governance strategies caused by international migration in the countries impacted by China’s Belt and Road initiative and their border areas.Buttressed by solid data mining and policy analysis, the title looks into the demographic trends of international migration in China and some Asian Belt and Road countries and stresses the urgency for more effective governance practices. Seeking to address the population security crisis triggered by the Initiative, the authors propose the idea of “multilateral population security governance”, grounded in the real-world challenges facing Belt and Road countries while also drawing on experiences of migration governance in western countries. As a new governance model, it calls for cross-border joint action and takes into consideration pertinent factors including economy, politics, culture, religion and commerce. Several case studies and comparative studies are offered in the chapters to illuminate the significance and effect of this cooperative mechanism.The book will be of interest to researchers and government officials interested in non-traditional security, international migration and formal demography as well as topics on population, resources and environment.
The Beltrami Equation: A Geometric Approach (Developments in Mathematics #26)
by Vladimir Gutlyanskii Vladimir Ryazanov Uri Srebro Eduard YakubovThis book is devoted to the Beltrami equations that play a significant role in Geometry, Analysis and Physics and, in particular, in the study of quasiconformal mappings and their generalizations, Riemann surfaces, Kleinian groups, Teichmuller spaces, Clifford analysis, meromorphic functions, low dimensional topology, holomorphic motions, complex dynamics, potential theory, electrostatics, magnetostatics, hydrodynamics and magneto-hydrodynamics.The purpose of this book is to present the recent developments in the theory of Beltrami equations; especially those concerning degenerate and alternating Beltrami equations. The authors study a wide circle of problems like convergence, existence, uniqueness, representation, removal of singularities, local distortion estimates and boundary behavior of solutions to the Beltrami equations. The monograph contains a number of new types of criteria in the given problems, particularly new integral conditions for the existence of regular solutions to the Beltrami equations that turned out to be not only sufficient but also necessary.The most important feature of this book concerns the unified geometric approach based on the modulus method that is effectively applied to solving the mentioned problems. Moreover, it is characteristic for the book application of many new concepts as strong ring solutions, tangent dilatations, weakly flat and strongly accessible boundaries, functions of finite mean oscillations and new integral conditions that make possible to realize a more deep and refined analysis of problems related to the Beltrami equations. Mastering and using these new tools also gives essential advantages for the reader in the research of modern problems in many other domains. Every mathematics graduate library should have a copy of this book.
BEM-based Finite Element Approaches on Polytopal Meshes (Lecture Notes in Computational Science and Engineering #130)
by Steffen WeißerThis book introduces readers to one of the first methods developed for the numerical treatment of boundary value problems on polygonal and polyhedral meshes, which it subsequently analyzes and applies in various scenarios. The BEM-based finite element approaches employs implicitly defined trial functions, which are treated locally by means of boundary integral equations. A detailed construction of high-order approximation spaces is discussed and applied to uniform, adaptive and anisotropic polytopal meshes. The main benefits of these general discretizations are the flexible handling they offer for meshes, and their natural incorporation of hanging nodes. This can especially be seen in adaptive finite element strategies and when anisotropic meshes are used. Moreover, this approach allows for problem-adapted approximation spaces as presented for convection-dominated diffusion equations. All theoretical results and considerations discussed in the book are verified and illustrated by several numerical examples and experiments. Given its scope, the book will be of interest to mathematicians in the field of boundary value problems, engineers with a (mathematical) background in finite element methods, and advanced graduate students.
A Benchmark Approach to Quantitative Finance (Springer Finance)
by Eckhard Platen David HeathA framework for financial market modeling, the benchmark approach extends beyond standard risk neutral pricing theory. It permits a unified treatment of portfolio optimization, derivative pricing, integrated risk management and insurance risk modeling. This book presents the necessary mathematical tools, followed by a thorough introduction to financial modeling under the benchmark approach, explaining various quantitative methods for the fair pricing and hedging of derivatives.
Benchmarking Economic Efficiency: Technical and Allocative Fundamentals (International Series in Operations Research & Management Science #315)
by Jesús T. Pastor Juan Aparicio José L. ZofíoThis book unifies and extends the definition and measurement of economic efficiency and its use as a real-life benchmarking technique for actual organizations. Analytically, the book relies on the economic theory of duality as guiding framework. Empirically, it shows how the alternative models can be implemented by way of Data Envelopment Analysis. An accompanying software programmed in the open-source Julia language is used to solve the models. The package is a self-contained set of functions that can be used for individual learning and instruction. The source code, associated documentation, and replication notebooks are available online. The book discusses the concept of economic efficiency at the firm level, comparing observed to optimal economic performance, and its decomposition according to technical and allocative criteria. Depending on the underlying technical efficiency measure, economic efficiency can be decomposed multiplicatively or additively. Part I of the book deals with the classic multiplicative approach that decomposes cost and revenue efficiency based on radial distance functions. Subsequently, the book examines how these partial approaches can be expanded to the notion of profitability efficiency, considering both the input and output dimensions of the firm, and relying on the generalized distance function for the measurement of technical efficiency. Part II is devoted to the recent additive framework related to the decomposition of economic inefficiency defined in terms of cost, revenue, and profit. The book presents economic models for the Russell and enhanced graph Russell measures, the weighted additive distance function, the directional distance function, the modified directional distance function, and the Hölder distance function. Each model is presented in a separate chapter. New approaches that qualify and generalize previous results are also introduced in the last chapters, including the reverse directional distance function and the general direct approach. The book concludes by highlighting the importance of benchmarking economic efficiency for all business stakeholders and recalling the main conclusions obtained from many years of research on this topic. The book offers different alternatives to measure economic efficiency based on a set of desirable properties and advises on the choice of specific economic efficiency models.
Benchmarking for Performance Evaluation: A Production Frontier Approach
by Subhash C. Ray Subal C. Kumbhakar Pami DuaThis book provides a detailed introduction to the theoretical and methodological foundations of production efficiency analysis using benchmarking. Two of the more popular methods of efficiency evaluation are Stochastic Frontier Analysis (SFA) and Data Envelopment Analysis (DEA), both of which are based on the concept of a production possibility set and its frontier. Depending on the assumed objectives of the decision-making unit, a Production, Cost, or Profit Frontier is constructed from observed data on input and output quantities and prices. While SFA uses different maximum likelihood estimation techniques to estimate a parametric frontier, DEA relies on mathematical programming to create a nonparametric frontier. Yet another alternative is the Convex Nonparametric Frontier, which is based on the assumed convexity of the production possibility set and creates a piecewise linear frontier consisting of a number of tangent hyper planes.Three of the papers in this volume provide a detailed and relatively easy to follow exposition of the underlying theory from neoclassical production economics and offer step-by-step instructions on the appropriate model to apply in different contexts and how to implement them. Of particular appeal are the instructions on (i) how to write the codes for different SFA models on STATA, (ii) how to write a VBA Macro for repetitive solution of the DEA problem for each production unit on Excel Solver, and (iii) how to write the codes for the Nonparametric Convex Frontier estimation. The three other papers in the volume are primarily theoretical and will be of interest to PhD students and researchers hoping to make methodological and conceptual contributions to the field of nonparametric efficiency analysis.
Benchmarking, Temporal Distribution, and Reconciliation Methods for Time Series (Lecture Notes in Statistics #186)
by Estela Bee Dagum Pierre A. CholetteTime series play a crucial role in modern economies at all levels of activity and are used by decision makers to plan for a better future. Before publication time series are subject to statistical adjustments and this is the first statistical book to systematically deal with the methods most often applied for such adjustments. Regression-based models are emphasized because of their clarity, ease of application, and superior results. Each topic is illustrated with real case examples. In order to facilitate understanding of their properties and limitations of the methods discussed a real data example is followed throughout the book.
Benchmarking with DEA, SFA, and R (International Series in Operations Research & Management Science #157)
by Peter Bogetoft Lars OttoThis book covers recent advances in efficiency evaluations, most notably Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) methods. It introduces the underlying theories, shows how to make the relevant calculations and discusses applications. The aim is to make the reader aware of the pros and cons of the different methods and to show how to use these methods in both standard and non-standard cases.Several software packages have been developed to solve some of the most common DEA and SFA models. This book relies on R, a free, open source software environment for statistical computing and graphics. This enables the reader to solve not only standard problems, but also many other problem variants. Using R, one can focus on understanding the context and developing a good model. One is not restricted to predefined model variants and to a one-size-fits-all approach. To facilitate the use of R, the authors have developed an R package called Benchmarking, which implements the main methods within both DEA and SFA.The book uses mathematical formulations of models and assumptions, but it de-emphasizes the formal proofs - in part by placing them in appendices -- or by referring to the original sources. Moreover, the book emphasizes the usage of the theories and the interpretations of the mathematical formulations. It includes a series of small examples, graphical illustrations, simple extensions and questions to think about. Also, it combines the formal models with less formal economic and organizational thinking. Last but not least it discusses some larger applications with significant practical impacts, including the design of benchmarking-based regulations of energy companies in different European countries, and the development of merger control programs for competition authorities.
Benchmarks and Hybrid Algorithms in Optimization and Applications (Springer Tracts in Nature-Inspired Computing)
by Xin-She YangThis book is specially focused on the latest developments and findings on hybrid algorithms and benchmarks in optimization and their applications in sciences, engineering, and industries. The book also provides some comprehensive reviews and surveys on implementations and coding aspects of benchmarks. The book is useful for Ph.D. students and researchers with a wide experience in the subject areas and also good reference for practitioners from academia and industrial applications.
The Bending Theory of Fully Nonlinear Beams
by Angelo Marcello Tarantino Luca Lanzoni Federico Oyedeji FalopeThis book presents the bending theory of hyperelastic beams in the context of finite elasticity. The main difficulties in addressing this issue are due to its fully nonlinear framework, which makes no assumptions regarding the size of the deformation and displacement fields. Despite the complexity of its mathematical formulation, the inflexion problem of nonlinear beams is frequently used in practice, and has numerous applications in the industrial, mechanical and civil sectors. Adopting a semi-inverse approach, the book formulates a three-dimensional kinematic model in which the longitudinal bending is accompanied by the transversal deformation of cross-sections. The results provided by the theoretical model are subsequently compared with those of numerical and experimental analyses. The numerical analysis is based on the finite element method (FEM), whereas a test equipment prototype was designed and fabricated for the experimental analysis. The experimental data was acquired using digital image correlation (DIC) instrumentation. These two further analyses serve to confirm the hypotheses underlying the theoretical model. In the book’s closing section, the analysis is generalized to the case of variable bending moment. The governing equations then take the form of a coupled system of three equations in integral form, which can be applied to a very wide class of equilibrium problems for nonlinear beams.
Benefit-Cost Analysis of Data Used to Allocate Funds (Lecture Notes in Statistics #3)
by Bruce SpencerThis monograph treats the question of determining how much to spend for the collection and analysis of public data. This difficult problem for government statisticians and policy-makers is likely to become even more pressing in the near future. The approach taken here is to estimate and compare the benefits and costs of alternative data programs. Since data are used in many ways, the benefits are hard to measure. The strategy I have adopted focuses on use of data to determine fund allocations, particularly in the General Revenue Sharing program. General Revenue Sharing is one of the largest allocation programs in the United States. That errors in population counts and other data cause sizable errors in allocation has been much publicized. Here we analyze whether the accuracy of the 1970 census of population and other data used by General Revenue Sharing should be improved. Of course it is too late to change the 1970 census program, but the method and techniques of analysis will apply to future data programs. In partic ular, benefit-cost analyses such as this are necessary for informed decisions about whether the expense of statistical programs is justi fied or not. For example, although a law authorizing a mid-decade census was enacted in 1976, there exists great doubt whether funds will be provided so a census can take place in 1985. (The President's Budget for 1981 allows no money for the mid-decade census, despite the Census Bureau's request for $1. 9 million for planning purposes.
Benefit/Cost-Driven Software Development: With Benefit Points and Size Points (Simula SpringerBriefs on Computing #8)
by Jo Erskine HannayThis open access book presents a set of basic techniques for estimating the benefit of IT development projects and portfolios. It also offers methods for monitoring how much of that estimated benefit is being achieved during projects. Readers can then use these benefit estimates together with cost estimates to create a benefit/cost index to help them decide which functionalities to send into construction and in what order. This allows them to focus on constructing the functionality that offers the best value for money at an early stage. Although benefits management involves a wide range of activities in addition to estimation and monitoring, the techniques in this book provides a clear guide to achieving what has always been the goal of project and portfolio stakeholders: developing systems that produce as much usefulness and value as possible for the money invested. The techniques can also help deal with vicarious motives and obstacles that prevent this happening. The book equips readers to recognize when a project budget should not be spent in full and resources be allocated elsewhere in a portfolio instead. It also provides development managers and upper management with common ground as a basis for making informed decisions.
Benefit-Risk Assessment in Pharmaceutical Research and Development
by Andreas Sashegyi James Felli Rebecca NoelMany practitioners in the pharmaceutical industry are still largely unfamiliar with benefit-risk assessment, despite its growing prominence in drug development and commercialization. Helping to alleviate this knowledge gap, Benefit-Risk Assessment in Pharmaceutical Research and Development provides a succinct overview of the key considerations rele
Benefit-Risk Assessment Methods in Medical Product Development: Bridging Qualitative and Quantitative Assessments (Chapman & Hall/CRC Biostatistics Series)
by Qi Jiang Weili HeGuides You on the Development and Implementation of B–R Evaluations Benefit–Risk Assessment Methods in Medical Product Development: Bridging Qualitative and Quantitative Assessments provides general guidance and case studies to aid practitioners in selecting specific benefit–risk (B–R) frameworks and quantitative methods. Leading experts from industry, regulatory agencies, and academia present practical examples, lessons learned, and best practices that illustrate how to conduct structured B–R assessment in clinical development and regulatory submission. The first section of the book discusses the role of B–R assessments in medicine development and regulation, the need for both a common B–R framework and patient input into B–R decisions, and future directions. The second section focuses on legislative and regulatory policy initiatives as well as decisions made at the U.S. FDA’s Center for Devices and Radiological Health. The third section examines key elements of B–R evaluations in a product’s life cycle, such as uncertainty evaluation and quantification, quantifying patient B–R trade-off preferences, ways to identify subgroups with the best B–R profiles, and data sources used to assist B–R assessment. The fourth section equips practitioners with tools to conduct B–R evaluations, including assessment methodologies, a quantitative joint modeling and joint evaluation framework, and several visualization tools. The final section presents a rich collection of case studies. With top specialists sharing their in-depth knowledge, thought-provoking considerations, and practical advice, this book offers comprehensive coverage of B–R evaluation methods, tools, and case studies. It gives practitioners a much-needed toolkit to develop and conduct their own B–R evaluations.
Benefit-Risk Assessment Methods in Medical Product Development: Bridging Qualitative and Quantitative Assessments (Chapman & Hall/CRC Biostatistics Series)
by Qi Jiang and Weili HeGuides You on the Development and Implementation of B–R Evaluations Benefit–Risk Assessment Methods in Medical Product Development: Bridging Qualitative and Quantitative Assessments provides general guidance and case studies to aid practitioners in selecting specific benefit–risk (B–R) frameworks and quantitative methods. Leading experts from industry, regulatory agencies, and academia present practical examples, lessons learned, and best practices that illustrate how to conduct structured B–R assessment in clinical development and regulatory submission. The first section of the book discusses the role of B–R assessments in medicine development and regulation, the need for both a common B–R framework and patient input into B–R decisions, and future directions. The second section focuses on legislative and regulatory policy initiatives as well as decisions made at the U.S. FDA’s Center for Devices and Radiological Health. The third section examines key elements of B–R evaluations in a product’s life cycle, such as uncertainty evaluation and quantification, quantifying patient B–R trade-off preferences, ways to identify subgroups with the best B–R profiles, and data sources used to assist B–R assessment. The fourth section equips practitioners with tools to conduct B–R evaluations, including assessment methodologies, a quantitative joint modeling and joint evaluation framework, and several visualization tools. The final section presents a rich collection of case studies. With top specialists sharing their in-depth knowledge, thought-provoking considerations, and practical advice, this book offers comprehensive coverage of B–R evaluation methods, tools, and case studies. It gives practitioners a much-needed toolkit to develop and conduct their own B–R evaluations.