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Bayes Linear Statistics
Bayes linear statistics is a subjectivist statistical methodology and framework. Traditional subjective Bayesian analysis is based upon fully specified probability distributions, which are very difficult to specify at the necessary level of detail. Bayes linear analysis attempts to solve this problem by developing theory and practise for using partially specified probability models. Bayes linear in its current form has been primarily developed by Michael Goldstein. Mathematically and philosophically it extends Bruno de Finetti's Operational Subjective approach to probability and statistics. Motivation Consider first a traditional Bayesian Analysis where you expect to shortly know ''D'' and you would like to know more about some other observable ''B''. In the traditional Bayesian approach it is required that every possible outcome is enumerated i.e. every possible outcome is the cross product of the partition of a set of ''B'' and ''D''. If represented on a computer where ''B' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Analysis
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Introduction to Bayes' rule Formal explanation Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem: P(H\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruno De Finetti
Bruno de Finetti (13 June 1906 – 20 July 1985) was an Italian probabilist statistician and actuary, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 "La prévision: ses lois logiques, ses sources subjectives," which discussed probability founded on the coherence of betting odds and the consequences of exchangeability. Life De Finetti was born in Innsbruck, Austria, and studied mathematics at Politecnico di Milano. He graduated in 1927 writing his thesis under the supervision of Giulio Vivanti. After graduation, he worked as an actuary and a statistician at ''Istituto Nazionale di Statistica'' ( National Institute of Statistics) in Rome and, from 1931, the Trieste insurance company Assicurazioni Generali. In 1936 he won a competition for Chair of Financial Mathematics and Statistics, but was not nominated due to a fascist law barring access to unmarried candidates; he was appointed as ordinary profess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherence (philosophical Gambling Strategy)
In a thought experiment proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability, an array of wagers is coherent precisely if it does not expose the wagerer to certain loss regardless of the outcomes of events on which they are wagering, even if their opponent makes the most judicious choices. Operational subjective probabilities as wagering odds One must set the price of a promise to pay $1 if John Smith wins tomorrow's election, and $0 otherwise. One knows that one's opponent will be able to choose either to buy such a promise from one at the price one has set, or require one to buy such a promise from them, still at the same price. In other words: Player A sets the odds, but Player B decides which side of the bet to take. The price one sets is the "operational subjective probability" that one assigns to the proposition on which one is betting. If one decides that John Smith is 12.5% likely to win—an arbitrary valuation—one might then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and Notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., ''X'' is a disjoint union of the subsets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said to exhaust or cover ''X''. See also collectively exhaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exchangeability
In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence ''X''1, ''X''2, ''X''3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences : X_1, X_2, X_3, X_4, X_5, X_6 \quad \text \quad X_3, X_6, X_1, X_5, X_2, X_4 both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models. Exchangeable sequences of random variables arise in cases of simple random sampling. Definition Formally, an exchangeable sequence of random variables is a finite or infinite sequence ''X''1, ''X''2, ''X''3, ... of random variables such that for any finite permutation σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Finetti's Theorem
In probability theory, de Finetti's theorem states that exchangeable observations are conditionally independent relative to some latent variable. An epistemic probability distribution could then be assigned to this variable. It is named in honor of Bruno de Finetti. For the special case of an exchangeable sequence of Bernoulli random variables it states that such a sequence is a "mixture" of sequences of independent and identically distributed (i.i.d.) Bernoulli random variables. A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of the indices. While the variables of the exchangeable sequence are not ''themselves'' independent, only exchangeable, there is an ''underlying'' family of i.i.d. random variables. That is, there are underlying, generally unobservable, quantities that are i.i.d. – exchangeable sequences are mixtures of i.i.d. sequences. Background A Bayesian statistician often seeks the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imprecise Probability
Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately. Imprecision is useful for dealing with expert elicitation, because: * People have a limited ability to determine their own subjective probabilities and might find that they can only provide an interval. * As an interval is compatible with a range of opinions, the analysis ought to be more convincing to a range of different people. Introduction Uncertainty is traditionally modelled by a probability distribution, as developed by Kolmogorov, Laplace, de Finetti, Ramsey, Cox, Lindley, and many others. However, this has not been unanimously accepted by scientists, statisticians, and probabilists: it has been argued that some modification or broadening of probabili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The Royal Statistical Society
The ''Journal of the Royal Statistical Society'' is a peer-reviewed scientific journal of statistics. It comprises three series and is published by Wiley for the Royal Statistical Society. History The Statistical Society of London was founded in 1834, but would not begin producing a journal for four years. From 1834 to 1837, members of the society would read the results of their studies to the other members, and some details were recorded in the proceedings. The first study reported to the society in 1834 was a simple survey of the occupations of people in Manchester, England. Conducted by going door-to-door and inquiring, the study revealed that the most common profession was mill-hands, followed closely by weavers. When founded, the membership of the Statistical Society of London overlapped almost completely with the statistical section of the British Association for the Advancement of Science. In 1837 a volume of ''Transactions of the Statistical Society of London'' were wri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AFM Smith
Sir Adrian Frederick Melhuish Smith, PRS (born 9 September 1946) is a British statistician who is chief executive of the Alan Turing Institute and president of the Royal Society. Early life and education Smith was born on 9 September 1946 in Dawlish. He was educated at Selwyn College, Cambridge, and University College London, where his PhD supervisor was Dennis Lindley. Career From 1977 until 1990, he was professor of statistics and head of department of mathematics at the University of Nottingham. He was subsequently at Imperial College, London, where he was head of the mathematics department. Smith is a former deputy vice-chancellor of the University of London and became vice-chancellor of the university on 1 September 2012. He stood down from the role in August 2018 to become the director of the Alan Turing Institute. Smith is a member of the governing body of the London Business School. He served on the Advisory Council for the Office for National Statistics from 1996 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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