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The article focuses on solving mathematical problems involving complex numbers and conic sections, including finding the curve on which a complex number 𝑧 z lies, evaluating limits involving quadratic roots, and determining parameters of a circle inscribed in an ellipse.

This article explores the use of Bayesian neural networks (BNNs) in mortality modeling, specifically in the context of South Africa. BNNs are gaining popularity due to their ability to incorporate prior assumptions and provide uncertainty quantification. The article outlines the steps involved in training a BNN, including assigning priors, defining likelihood models, and using inference methods such as Markov chain Monte Carlo (MCMC) or variational inference. The results show that the BNN performs well in capturing trends and making predictions compared to traditional models, but further exploration and testing are recommended. The authors acknowledge the expertise of professionals in the field. [Extracted from the article]

In the book "The Art of Uncertainty: How to navigate chance, ignorance, risk and luck," statistician David Spiegelhalter explores the concept of uncertainty and how it affects our lives. He explains that the chance of two non-twin siblings sharing a birthday is still 1 in 365, despite the individual chance of being born on a particular day being the same. The book delves into statistical concepts and provides examples to help readers understand everyday uncertainties. Spiegelhalter also discusses the role of luck and how it can be influenced by traits such as taking advantage of opportunities and having positive expectations. While the book doesn't offer a formula for manipulating luck, it provides insights into understanding and navigating uncertainty. [Extracted from the article]

One of the biggest challenges facing any statistician is communicating their work to people who don't eat, sleep and breathe data to the extent they do. Even the most thorough courses in statistics can leave students – be they undergraduates or PhD candidates – at a loss as to how to share their knowledge effectively and persuasively, be it via a work presentation or TED talk, personal blog or magazine article, newsletter, X thread, or even just a conversation with a boss or friend. So how do we help the non-expert get their head around complex statistics, and at the same time find our voices as messengers? In this special feature, we ask three of the best stats communicators working today to tell us what they do, and how and why they do it. Because rarely does the data speak for itself. What happens when you combine creativity with data nous? Daniel Parris tells Hayley Bennett about his unorthodox journey into the statistics world, and why he started his must-read newsletter for numerate culture-vultures Want to create more memorable data viz? Take your time, go for a walk and remember it's OK to suck at the start, Alli Torban tells Hayley Bennett As a science writer for British newspapers, author of popular books on statistical matters and twice winner of the Royal Statistical Society's award for statistical excellence in journalism, Tom Chivers is versed in demystifying complex concepts for mainstream readers. His latest book explains Bayes' theorem to the non-statistician. [ABSTRACT FROM AUTHOR]

The article focuses on a series of mathematical problems spanning various topics, including curves intersection, quadratic equations, definite integration, matrix determinants, geometry, differential equations, and probability. It is discussed that these problems challenge readers to apply algebraic techniques, geometric reasoning, calculus, and probability theory to find solutions.

New Scientist offers a range of resources for library patrons conducting research. They are hosting an event called New Scientist Live, where science writer Tom Chivers will discuss Bayes's theorem. They also offer a tour of Niels Bohr's Copenhagen, exploring the birthplace of quantum physics and the historical city of Roskilde. In their podcast, they interview astrophysicist Lisa Kaltenegger about her new book on exoplanets. The European Space Agency's EarthCARE satellite launch can be watched on their YouTube channel, providing insight into cloud formation and climate change. They also discuss the recent discovery of ancient rock art in South America in their newsletter, challenging previous assumptions about the distribution of such art. Lastly, they have an Essential Guide on space exploration available for reading or purchase. [Extracted from the article]

The article focuses on the trend analysis of Central Board of Secondary Education Board questions from 2014 to 2023, highlighting topics such as Mathematics National Council of Educational Research and Training topics, Inverse Trigonometric Functions, Relations and Functions, Matrices, and more. It presents the graphical analysis identifies significant topics, accompanied by High Definition images, while a pie chart analysis illustrates the total weightage of each topic.

The article focuses on various mathematical problems and syllabus related to class 12 Olympiads, including relations, trigonometric functions, matrices, determinants, and differential equations. Topics include determining the properties of given relations, evaluating trigonometric expressions, finding the powers of matrices, expanding determinants, and analyzing functions' continuity and differentiation.

The article discusses chapter-wise practice questions for Class XII Central Board of Secondary Education (CBSE) exams, following the updated syllabus for 2023-2024, specifically covering linear programming and probability in mathematics topics in the "CBSE warm-up!" section.

The article focuses on the December 2023 edition of "Mathematics Today," covering various mathematical problems and questions across different sections, including Logical Reasoning, Mathematical Reasoning and Everyday Mathematics. It delves into topics such as sets, functions, limits, derivatives, probability, differential equations, matrices, determinants and more, challenging students with diverse mathematical problems and concepts.

This document titled "Elsewhere on New Scientist" provides a brief overview of various topics covered in the New Scientist publication. It mentions a discussion on Bayes's theorem and its applications in medical testing and artificial intelligence. The document also highlights a tour of ancient caves in Northern Spain, a podcast episode exploring the shape of galaxies, a video showcasing the ornithological collection at the Natural History Museum in Tring, and a newsletter discussing the impact of housebuilding on wildlife. Additionally, it mentions an essential guide on space exploration and includes some colorful photographs related to the topics mentioned. [Extracted from the article]

A quiz with mathematical questions for Olympiads for class XII students is presented.

The article presents some of the important topics for Central Board of Secondary Education board exams, which includes relations and functions; inverse trigonometric functions; and matrices and determinants.

A quiz with mathematical questions is presented.

A recent study conducted at Moi University School of Medicine in Western Kenya aimed to determine the correlation between colposcopic and final histopathologic results among patients undergoing evaluation for cervical dysplasia. The study included 164 female participants, with findings indicating a relatively strong correlation between colposcopy and biopsy results. However, colposcopy showed lower specificity compared to histopathology. This research provides valuable insights into the evaluation of cervical dysplasia in clinical settings. [Extracted from the article]

This article discusses the increasing popularity of Bayesian statistics in the field of psychology. It highlights the advantages of Bayesian methods and aims to introduce and explain these methods to psychology researchers who may be unfamiliar with them. The tutorial compares Bayesian and frequentist approaches, defines key terms, and provides practical illustrations. It also includes a psychological research example with real data, analyzing it using both frequentist and Bayesian approaches. The article emphasizes best practices for interpretation and reporting and provides R code and supplementary materials. [Extracted from the article]

Researchers at Juntendo University Nerima Hospital in Tokyo, Japan, have conducted a study on chest pain and its diagnosis. The study found that acute coronary syndrome (ACS) is associated with high mortality rates, and the missed diagnosis rate is higher than the desired rate. The researchers recommend that medical interviews focus on the character and location of chest pain, and electrocardiograms (ECG) should be recorded within 10 minutes of presentation. Troponin levels should be examined in all patients with suspected ischemic heart disease. The study concludes that patients should be risk stratified based on serial measurements of cardiac troponin levels to determine the appropriate time for a definitive diagnosis. [Extracted from the article]

A new report discusses research findings on Pain and Central Nervous System, specifically focusing on the use of plasma phosphorylated tau (p-tau)217 as a biomarker for Alzheimer's disease diagnosis. The researchers propose a strategy based on Bayes' theorem and test it in real-life memory clinics. The Bayesian approach demonstrated high accuracy and outperformed other models for AD diagnosis. The research concludes that the Bayesian strategy offers an effective and flexible approach to address the limitations of plasma p-tau217 in clinical practice. The study was conducted at King Chulalongkorn Memorial Hospital in Bangkok, Thailand, and was supported by the Health Systems Research Institute. [Extracted from the article]

This article discusses the increasing popularity of Bayesian statistics in the field of psychology. It highlights the need for researchers to become familiar with Bayesian methods and introduces Bayesian analysis as a powerful tool. The tutorial compares Bayesian and frequentist approaches, explains fundamental terms, and provides practical illustrations. It also includes a psychological research example with real data, analyzing it using both frequentist and Bayesian approaches. The article emphasizes best practices for interpretation and reporting and maintains an accessible and user-friendly language throughout. Please note that this preprint has not been peer-reviewed. [Extracted from the article]

The article discusses a puzzle presented to Sherlock Holmes by his brother Mycroft. In the puzzle, Holmes is asked to determine the numbers on two spinners that, when used in a dice game, produce the same probabilities as two standard six-faced dice. The article also includes the solution to a previous puzzle, "The case of the Green Cane of Kabul," which involved the use of Bayes' theorem. The article concludes with an apology for a mistake made in referencing a Sherlock Holmes story. [Extracted from the article]

A recent scoping review conducted by researchers at Miguel Hernandez University in Elche, Spain, explores the use of Bayesian networks (BNs) in disease diagnosis and prognosis. Bayesian networks are probabilistic graphical models that can depict relationships between variables and have gained prominence in the field of health sciences. The review analyzed 60 relevant papers and found that BNs have shown promise in various medical areas, with an average area under the curve (AUC) exceeding 75%. The study concludes that BNs offer a viable solution for addressing complex medical challenges and highlights the need for further research to optimize their capabilities. [Extracted from the article]

The article discusses the increasing use of Bayesian inference in a variety of contexts. Scientific consulting firm Metron used Bayesian methods to locate an Air France jetliner that crashed into the Atlantic Ocean in June 2009. Bayesian inference helped political blogger Nate Silver correctly predict how all 50 U.S. states would vote in the November 2012 presidential election. The author describes the basics of Bayes Law, how it differs from frequentist inference, and comments on how Bayesian methods have benefited from increases in computational power.

Social scientists have developed dozens of different ways to quantify urban segregation with numerical statistics. Behind the scenes, some of these measurements are defined using multivariable functions that model the distribution of various groups in a geographical region. Using Bayes' theorem and kernel density estimation, we describe how to summarize population data in terms of smooth, two-dimensional surfaces. These surfaces give us ways of identifying neighborhood boundaries and visualizing segregation patterns. We also propose two new segregation measures using some familiar tools from third-semester calculus. Applications to United States census data are included. [ABSTRACT FROM AUTHOR]

• A robust generative classifier based on variational auto-encoders • The classifier is more robust against transfer-based attacks. • Minimize the KL divergence to get the evidence lower bound • Consider a Gaussian mixture prior on the latent variable. Deep neural networks (DNNs) are vulnerable to adversarial examples. Even under black-box setting that is without access to the target model, transfer-based attacks can easily fool the DNNs. To alleviate this problem, we propose a robust classification model against transfer attacks based on the framework of variational Auto-Encoders (VAEs) which are probabilistic generative models and have been successfully used to a large mount of tasks. Specifically, our model simulates the data generative process with several multivariate Gaussian distributions and DNNs: (1) We assume that the latent embedding generated by an encoder (a DNN) of each category corresponds to a multivariate Gaussian distribution. (2) A decoder (a DNN) is proposed to decodes the latent embedding into an observable. (3) Theoretical analysis illustrates that our model can predict data's labels by maximizing the lower bound on the log-likelihood for each category utilizing Bayes' theorem with excellent robustness against transfer attacks. Inference in our model is done in a variational way so the Stochastic Gradient Variational Bayes (SGVB) estimator and reparamerization trick can be utilized to optimize the evidence lower bound (ELBO). The experiments with quantitative comparisons show that our approach reaches state-of-the-art with significantly better robustness. [ABSTRACT FROM AUTHOR]

In this article, a general theoretical framework is developed for the state estimation problem of stochastic time-varying Boolean networks (STVBNs). The STVBN consists of a system model describing the evolution of the Boolean states and a model relating the noisy measurements to the Boolean states. Both the process noise and the measurement noise are characterized by sequences of mutually independent Bernoulli distributed stochastic variables taking values of 1 or 0, which imply that the state/measurement variables may be flipped with certain probabilities. First, an algebraic representation of the STVBNs is derived based on the semitensor product. Then, based on Bayes’ theorem, a recursive matrix-based algorithm is obtained to calculate the one-step prediction and estimation of the forward–backward state probability distribution vectors. Owing to the Boolean nature of the state variables, the Boolean Bayesian filter is designed that can be utilized to provide the minimum MSE state estimate for the STVBNs. The fixed-interval smoothing filter is also obtained by resorting to the forward–backward technique. Finally, a simulation experiment is carried out for the context estimation problem of the $p53$ - $MDM2$ negative-feedback gene regulatory network. [ABSTRACT FROM AUTHOR]

Asness and Brown acknowledged that having a stateful model might improve the decision-making power of the AB model; specifically, that knowing if the puck is in the offensive zone, neutral zone, or defensive zone would vary their recommendation. When one team has pulled the goalie, the scoring rate becomes 1.97% per 10-second interval for the team with the sixth attacker, and 4.3% for the team that keeps its goalie. If the time is between 330 seconds and less than or equals to 550 seconds, the coach should pull the goalie when the puck is in the offensive zone, and put the goalie back when the puck is in the neutral or defensive zone. [Extracted from the article]

• Supervised feature selection (FS) approach for high dimensional datasets with small sample size. • Formulated FS as a set-covering problem by defining instance votes to features. • Achieved lower misclassification rates on average as compared to MI based methods. • Selected features were robust to minor data variations. • Stopping criterion automatically determines size of selected feature subsets. In this work, we address the problem of supervised feature selection (FS) for high-dimensional datasets with a small number of instances. Here, we propose a novel heuristic FS approach, Conditional Priority Coverage Maximization (CPCM) which seeks to leverage the local information provided by the small set of instances. We define the vote assigned by an instance to a feature as the local relevance of the latter. Also, we show that the proposed voting scheme is asymptotically related to the Bayes' decision rule for minimum risk classification. Next, we exploit the instance votes for feature selection by posing it as a set-covering problem — we seek to select a subset of features such that they can together cover the instances. This approach avoids the selection of redundant features, while selecting relevant ones. In addition, we formulate the stopping criterion to select a compact subset of features. Through experiments on synthetic and real datasets, we demonstrated that CPCM outperforms other graph based FS techniques and state-of-the-art FS approaches employing mutual information (MI). Further, we evaluated the stability of CPCM to minor variations in the training data and found it to be reasonably robust. [ABSTRACT FROM AUTHOR]

A recent report from Colorado State University discusses the limitations of assuming Gaussian error statistics in data assimilation, which is the technique of combining model predictions with observations. The researchers propose a method that allows for non-Gaussian error distributions, including skewness and kurtosis, and permits physical bounds for variables. They apply this framework to a 3D atmospheric toy model and find improved performance compared to an all-Gaussian technique. This research provides a general framework for non-Gaussian data assimilation and has the potential to improve forecasting accuracy. [Extracted from the article]

The authors maintain that mathematics lessons should be planned around "serious or provocative" issues and that "mathematics is a means to challenge the status quo." Not only that - in fact, uncertainty about the values of outcomes in an indefinite future, and about the probabilities of never-before-realized events, render expected value utterly unsuitable for a "moral mathematics." "What place does social justice teaching and learning have in a math class?". [Extracted from the article]

Thermal conductivity is a major influencing factor on subsurface conductive heat transport and resulting temperature distribution, which in turn is a key parameter in basin modeling. Basin modeling studies commonly use representative literature values of thermal conductivity despite their impact on modeling results. We introduce a workflow for quantifying the effect of uncertain thermal conductivity on subsurface temperature distribution and thus on basin modeling results and test this workflow on a twodimensional generic model from the Nordkapp Basin; a prior ensemble of possible models is conditioned according to Bayes' theorem to incorporate prior knowledge of temperature data. This conditional probability yields a posterior ensemble of temperature fields with a significantly reduced standard deviation. To verify our approach, we use five characteristic scenarios from the posterior ensemble for transient petroleum systems modeling. How considering uncertain thermal conductivity affects variance in hydrocarbon generation is assessed by modeling corresponding vitrinite reflectances (R0). Temperature uncertainty increases with depth. It also increases with increasing offset from the salt diapirs, which can be associated with a large lateral heat-flow component in the complex tectonic environment of the Nordkapp Basin. The introduced workflow can reduce temperature uncertainty significantly, especially in regions with high prior uncertainty. The R0 is very sensitive to changes in thermal conductivity because the onset depth of the gas window in the Nordkapp Basin may vary by up to 800 m (2600 ft) within the 95% confidence interval. This demonstrates the importance of quantification of the uncertainty in thermal conductivity on thermal basin modeling. [ABSTRACT FROM AUTHOR]

3 IN 2012, PROFESSOR ARA NORENZAYAN at the University of British Columbia claimed to have evidence that looking at an image of Rodin's sculpture The Thinker could make people less religious. Fisher claimed significance testing was a universal tool for scientific inference, "common to all experimentation", a claim that seems borne out by its widespread use across all disciplines. In short, significance testing would have told our hypothetical patient that she probably has cancer and would have wrongfully convicted Sally Clark. Meadow came up with his figure of 1 in 73 million by applying some adjustments to the observed incidence rate of SIDS (about 1 in 1,300) to account for what was known about the Clark family: They were nonsmokers with steady jobs and Sally was over the age of 26. [Extracted from the article]

Learning a hidden Markov model (HMM) is typically based on the computation of a likelihood which is intractable due to a summation over all possible combinations of states and mixture components. This estimation is often tackled by a maximization strategy, which is known as the Baum–Welch algorithm. However, some drawbacks of this approach have led to the consideration of Bayesian methods that add a prior over the parameters in order to work with the posterior probability and the marginal likelihood. These approaches can lead to good models but to the cost of extremely long computations (e.g., Markov Chain Monte Carlo). More recently, variational Bayesian frameworks have been proposed as a Bayesian alternative that keeps the computation tractable and the approximation tight. It relies on the introduction of a prior over the parameters to be learned and on an approximation of the true posterior distribution. After proving good standing in the case of finite mixture models and discrete and Gaussian HMMs, we propose here to derive the equations of the variational learning of the Dirichlet mixture-based HMM, and to extend it to the generalized Dirichlet. The latter case presents several properties that make the estimation more accurate. We prove the validity of this approach within the context of unusual event detection in public areas using the University of California San Diego data sets. HMMs are trained over normal video sequences using the typical Baum–Welch approach versus the variational one. The variational learning leads to more accurate models for the detection and localization of anomaly, and the general HMM approach is shown to be versatile enough to handle the detection of various synthetically generated tampering events. [ABSTRACT FROM AUTHOR]

Abstract A cointegrated linear process in Bayes Hilbert space is isomorphic to a cointegrated linear process in a Hilbert space of centered square-integrable real functions. We illustrate the use of this isomorphism for modeling nonstationary time series of probability densities. [ABSTRACT FROM AUTHOR]

Abstract Based on the notion of information and knowledge granularity, sequential three-way decisions can be interpreted and implemented as a multilevel trisecting-and-acting framework which leads to a faster decision-making with the less cost of decision process. At each level, objects are divided into the triplet, consisting of three pair-wise disjoint regions. By considering different combinations of regions, the uncertain objects should be further investigated at the next level due to insufficient information. In fact, sequential strategy is an effective idea to deal with multi-class problem in granular computing. On the basis of this recognition, we focus on a sequential three-way approach to multi-class decision in this paper. To solve the issues of unequal cost and rule conflict simultaneously, we present a novel matrix-based multi-class decision method based on Bayesian decision procedure. Subsequently, a sequence of the proposed multi-class models is conducted under a multilevel granular structure, and objects are assigned into one certain decision class gradually from the lower levels to the higher levels via a top-down manner. Finally, the results of empirical study involving six multi-class data sets on UCI are reported to validate the feasibility and effectiveness of our proposal. [ABSTRACT FROM AUTHOR]

Focuses on Bayes' theorem for decision making. Background information on Reverend Thomas Bayes; Comparison with year ago performance; Description of Bayesian thinking; Development into Laplacean statistical thinking; Case of the drugs t-PA and streptokinase; Implications for changing attitudes.

Focuses on the recognition of the Bayes theorem in the age of high technology. Fundamental level of the theorem; Problems of probability in the field of science; Involvement of the Bayesian inference in extrasensory perception; Ability of the Bayes theorem to reveal true significance of new findings. INSET: Bayes in action..

Ord, by contrast, estimates the risk of extinction by asteroid at one in a million, though he does note a considerable degree of uncertainty. Ord's book, for example, contains a table of potential extinction events and his personal estimates of their probability. In 2020, Oxford-based philosopher Toby Ord published a book called The Precipice about the risk of human extinction. [Extracted from the article]

A method for sequential Bayesian inference of the static parameters of a dynamic state space model is proposed. The method is able to use any valid approximation to the filtering and prediction densities of the state process. It computes the posterior distribution of the static parameters on a discrete grid that tracks the support dynamically. For inference of the state process, the Kalman filter and its extensions as well as cubature filtering have been used. It is illustrated with several examples including the stochastic volatility model and the challenging Kitagawa model and is compared to both online and offline methods. It is shown to provide a good trade off between speed and performance. [ABSTRACT FROM AUTHOR]

How Bayesian inference can decode movement intentions and control the next generation of powered prostheses. By Mike Wininger and Reva Johnson [ABSTRACT FROM AUTHOR]

Two procedures for supervised classification are proposed. These are based on data depth and focus on the centre of each class. The classifiers add either a depth or a depth rank term to the objective function of the Bayes classifier. The cost of misclassifying a point depends not only on a class where it belongs, but also on its centrality with respect to this class. The classification of points that are more central is enforced while outliers are downweighted. The proposed objective function can also be used to evaluate the performance of other classifiers instead of the usual average misclassification rate. Use of the depth function increases robustness of the new procedures against the large inclusion of contaminated data that often impede the Bayes classifier. Properties of the new methods are investigated and compared with those of the Bayes classifier. Theoretical results are derived for elliptically symmetric distributions, while comparison for non-symmetric distributions is conducted by means of a simulation study. Comparisons are conducted for both theoretical classifiers and their empirical counterparts. The performance of the newly proposed classifiers is also compared to the performance of several standard methods in some real life situations. [ABSTRACT FROM AUTHOR]

Traditionally, algorithms that learn the structure of Bayesian Networks either start from an empty graph and add edges to it bit by bit or add/ remove/reverse edges in a randomly initialized graph. In both cases, the search space is constituted of all the nodes and edges connecting them. Searching within such a vast scope is a hard task, which gets worse in the presence of a dataset with missing values. However, it may be the case that an initial structure already exists and to make it reflect the set of examples it would be required to modify only a subset of the graph. Thus, instead of searching through the entire space of possible connections between the nodes, the problem could be reduced to selecting a subset of the edges and revising them. In this work, we present a novel algorithm for refining the structure of Bayesian networks from incomplete data, named BaBReN (Bayes Ball for Revising Networks). BaBReN has as ultimate goal to improve the inference value of the class variable. Thus, the algorithm tries to solve classification issues by proposing local modifications to the edges connecting the nodes that influence the erroneous classification. The Bayes Ball algorithm – based on the d-separation criteria – is responsible for selecting those relevant nodes. By focusing only on the influential nodes, BaBReN is executed independently of the number of variables in the domain. BaBReN is compared to a constraint-based algorithm (GS), a hybrid one (MMHC) and a score-based one (SEM with GHC), presenting better or competitive results regarding time and classification score. [ABSTRACT FROM AUTHOR]

Multidimensional Bayesian network classifiers have gained popularity over the last few years due to their expressive power and their intuitive graphical representation. A drawback of this approach is that their use to perform multidimensional classification, a generalization of multi-label classification, can be very computationally demanding when there are a large number of class variables. Thus, a key challenge in this field is to ensure the tractability of these models during the learning process. In this paper, we show how information about the most common queries of multidimensional Bayesian network classifiers affects the complexity of these models. We provide upper bounds for the complexity of the most probable explanations and marginals of class variables conditioned to an instantiation of all feature variables. We use these bounds to propose efficient strategies for bounding the complexity of multidimensional Bayesian network classifiers during the learning process, and provide a simple learning method with an order-based search that guarantees the tractability of the returned models. Experimental results show that our approach is competitive with other methods in the state of the art and also ensures the tractability of the learned models. [ABSTRACT FROM AUTHOR]

We consider a new kernel rule of classification for spatially dependent data. This nonparametric rule allows for the classification of missing observations. We investigate the consistency of this classification rule and we propose a method for bandwidth selection. [ABSTRACT FROM AUTHOR]