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Bayesian Statistics, Theorem
Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a follower of these methods. A number of things have been named after Thomas Bayes, including: Bayes *Bayes action *Bayes Business School * Bayes classifier * Bayes discriminability index *Bayes error rate *Bayes estimator *Bayes factor * Bayes Impact *Bayes linear statistics * Bayes prior * Bayes' theorem / Bayes-Price theorem -- sometimes called Bayes' rule or Bayesian updating. *Empirical Bayes method *Evidence under Bayes theorem *Hierarchical Bayes model *Laplace–Bayes estimator *Naive Bayes classifier * Random naive Bayes Bayesian *Approximate Bayesian computation *Bayesian average *Bayesian Analysis (journal) *Bayesian approaches to brain function * Bayesian bootstrap *Bayesian control rule *Bayesian cognitive science *Bayesian eco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Bayes
Thomas Bayes ( ; 1701 7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem. Bayes never published what would become his most famous accomplishment; his notes were edited and published posthumously by Richard Price. Biography Thomas Bayes was the son of London Presbyterian minister Joshua Bayes, and was possibly born in Hertfordshire. He came from a prominent Nonconformist (Protestantism), nonconformist family from Sheffield. In 1719, he enrolled at the University of Edinburgh to study logic and theology. On his return around 1722, he assisted his father at the latter's chapel in London before moving to Royal Tunbridge Wells, Tunbridge Wells, Kent, around 1734. There he was minister of the Mount Sion Chapel, until 1752. He is known to have published two works in his lifetime, one theological and one mathematical: #''Divine Benevolence, or an Attempt to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Naive Bayes Classifier
In statistics, naive Bayes classifiers are a family of simple "probabilistic classifiers" based on applying Bayes' theorem with strong (naive) independence assumptions between the features (see Bayes classifier). They are among the simplest Bayesian network models, but coupled with kernel density estimation, they can achieve high accuracy levels. Naive Bayes classifiers are highly scalable, requiring a number of parameters linear in the number of variables (features/predictors) in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression, which takes linear time, rather than by expensive iterative approximation as used for many other types of classifiers. In the statistics literature, naive Bayes models are known under a variety of names, including simple Bayes and independence Bayes. All these names reference the use of Bayes' theorem in the classifier's decision rule, but naive Bayes is not (necessarily) a Bayesian method. Introductio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Experimental Design
Bayesian experimental design provides a general probability-theoretical framework from which other theories on experimental design can be derived. It is based on Bayesian inference to interpret the observations/data acquired during the experiment. This allows accounting for both any prior knowledge on the parameters to be determined as well as uncertainties in observations. The theory of Bayesian experimental design is to a certain extent based on the theory for making optimal decisions under uncertainty. The aim when designing an experiment is to maximize the expected utility of the experiment outcome. The utility is most commonly defined in terms of a measure of the accuracy of the information provided by the experiment (e.g. the Shannon information or the negative of the variance), but may also involve factors such as the financial cost of performing the experiment. What will be the optimal experiment design depends on the particular utility criterion chosen. Relations to more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loss Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Epistemology
Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. One advantage of its formal method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision. It is based on the idea that beliefs can be interpreted as Subjective probability, subjective probabilities. As such, they are subject to the laws of probability theory, which act as the norms of rationality. These norms can be divided into static constraints, governing the rationality of beliefs at any moment, and dynamic constraints, governing how rational agents should change their beliefs upon receiving new evidence. The most characteristic Bayesian expression of these principles is found in the form of Dutch books, which illustrate irrationality in agents through a series of bets that lead to a loss for the agent no matter which of the probabilistic events occurs. Bayesian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Efficiency
Bayesian efficiency is an analog of Pareto efficiency for situations in which there is incomplete information.Palfrey, Thomas R.; Srivastava, Sanjay; Postlewaite, A. (1993) Bayesian Implementation.' Pg. 13-14. Under Pareto efficiency, an allocation of a resource is Pareto efficient if there is no other allocation of that resource that makes no one worse off while making some agents strictly better off. A limitation with the concept of Pareto efficiency is that it assumes that knowledge about other market participants is available to all participants, in that every player knows the payoffs and strategies available to other players so as to have complete information. Often, the players have types that are hidden from the other player. Overview The lack of complete information raises a question of when the efficiency calculation should be made. Should the efficiency check be made at the ex ante stage before the agent sees their types, at the interim stage after the agent sees their t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Econometrics
Bayesian econometrics is a branch of econometrics which applies Bayesian principles to economic modelling. Bayesianism is based on a degree-of-belief interpretation of probability, as opposed to a relative-frequency interpretation. The Bayesian principle relies on Bayes' theorem which states that the probability of B conditional on A is the ratio of joint probability of A and B divided by probability of B. Bayesian econometricians assume that coefficients in the model have prior distributions. This approach was first propagated by Arnold Zellner. Basics Subjective probabilities have to satisfy the standard axioms of probability theory if one wishes to avoid losing a bet regardless of the outcome. Before the data is observed, the parameter \theta is regarded as an unknown quantity and thus random variable, which is assigned a prior distribution \pi(\theta) with 0 \leq \theta \leq 1. Bayesian analysis concentrates on the inference of the posterior distribution \pi(\theta, y) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Cognitive Science
Bayesian cognitive science, also known as computational cognitive science, is an approach to cognitive science concerned with the rational analysis of cognition through the use of Bayesian inference and cognitive modeling. The term "computational" refers to the computational level of analysis as put forth by David Marr. This work often consists of testing the hypothesis that cognitive systems behave like rational Bayesian agents in particular types of tasks. Past work has applied this idea to categorization, language, motor control, sequence learning, reinforcement learning and theory of mind. At other times, Bayesian rationality is ''assumed'', and the goal is to infer the knowledge that agents have, and the mental representations that they use. It is important to contrast this with the ordinary use of Bayesian inference in cognitive science, which is independent of rational modeling (see e.gMichael Lee's work. See also * Active inference * Bayesian approaches to brai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Control Rule
Thompson sampling, named after William R. Thompson, is a heuristic for choosing actions that addresses the exploration-exploitation dilemma in the multi-armed bandit In probability theory and machine learning, the multi-armed bandit problem (sometimes called the ''K''- or ''N''-armed bandit problem) is a problem in which a fixed limited set of resources must be allocated between competing (alternative) choices ... problem. It consists of choosing the action that maximizes the expected reward with respect to a randomly drawn belief. Description Consider a set of contexts \mathcal, a set of actions \mathcal, and rewards in \mathbb. In each round, the player obtains a context x \in \mathcal, plays an action a \in \mathcal and receives a reward r \in \mathbb following a distribution that depends on the context and the issued action. The aim of the player is to play actions such as to maximize the cumulative rewards. The elements of Thompson sampling are as follows: # a likeliho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bootstrapping (statistics)
Bootstrapping is any test or metric that uses random sampling with replacement (e.g. mimicking the sampling process), and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, etc.) to sample estimates.software This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Bootstrapping estimates the properties of an (such as its ) by measuring those properties when sampling from an approximating distribution. One standard choice for an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Approaches To Brain Function
Bayesian approaches to brain function investigate the capacity of the nervous system to operate in situations of uncertainty in a fashion that is close to the optimal prescribed by Bayesian statistics. This term is used in behavioural sciences and neuroscience and studies associated with this term often strive to explain the brain's cognitive abilities based on statistical principles. It is frequently assumed that the nervous system maintains internal probabilistic models that are updated by neural processing of sensory information using methods approximating those of Bayesian probability. Origins This field of study has its historical roots in numerous disciplines including machine learning, experimental psychology and Bayesian statistics. As early as the 1860s, with the work of Hermann Helmholtz in experimental psychology the brain's ability to extract perceptual information from sensory data was modeled in terms of probabilistic estimation. The basic idea is that the nervous sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Analysis (journal)
''Bayesian Analysis'' is an open-access peer-reviewed scientific journal covering theoretical and applied aspects of Bayesian methods. It is published by the International Society for Bayesian Analysis and is hosted at the Project Euclid web site. ''Bayesian Analysis'' is abstracted and indexed by Science Citation Index Expanded. According to the ''Journal Citation Reports'', the journal has a 2011 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 1.650. References External links * Statistics journals Open access journals Publications established in 2006 English-language journals Quarterly journals Academic journals published by international learned and professional societies {{statistics-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
