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Bayesian Probability
Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability. Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a prior probability. This, in turn, is then updated to a posterior probability in the light of new, relevant data (evidence). The Bayesian interpretation provides a standard set of procedures and form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Interpretations
The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory. There are two broad categories The taxonomy of probability interpretations given here is similar to that of the longer and more complete Interpretations of Probability article in the online Stanford Encyclopedia of Philosophy. References to that article include a parenthetic section number where appropriate. A partial outline of that article: * Section 2: Criteria of adequacy for the interpretations of probability * Section 3: ** 3.1 Classical Probability ** 3.2 Logical Probability ** 3.3 Subjective Probability ** 3.4 Frequency Interpretations ** 3.5 Propensity Interpretations " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayes' Theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately (by conditioning it on their age) than simply assuming that the individual is typical of the population as a whole. One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in the theorem may have different probability interpretations. With Bayesian probability interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of related evidence. Bayesian inference is fundamental to Bayesia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reliability (statistics)
In statistics and psychometrics, reliability is the overall consistency of a measure. A measure is said to have a high reliability if it produces similar results under consistent conditions:"It is the characteristic of a set of test scores that relates to the amount of random error from the measurement process that might be embedded in the scores. Scores that are highly reliable are precise, reproducible, and consistent from one testing occasion to another. That is, if the testing process were repeated with a group of test takers, essentially the same results would be obtained. Various kinds of reliability coefficients, with values ranging between 0.00 (much error) and 1.00 (no error), are usually used to indicate the amount of error in the scores." For example, measurements of people's height and weight are often extremely reliable.The Marketing Accountability Standards Board (MASB) endorses this definition as part of its ongoinCommon Language: Marketing Activities and Metrics P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Celestial Mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data. History Modern analytic celestial mechanics started with Isaac Newton's Principia of 1687. The name "celestial mechanics" is more recent than that. Newton wrote that the field should be called "rational mechanics." The term "dynamics" came in a little later with Gottfried Leibniz, and over a century after Newton, Pierre-Simon Laplace introduced the term "celestial mechanics." Prior to Kepler there was little connection between exact, quantitative prediction of planetary positions, using geometrical or arithmetical techniques, and contemporary discussions of the physical causes of the planets' motion. Johannes Kepler Johannes Kepler (1571–1630) was the first to closely integrate the predictive geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Trial
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his ''Ars Conjectandi'' (1713). The mathematical formalisation of the Bernoulli trial is known as the Bernoulli process. This article offers an elementary introduction to the concept, whereas the article on the Bernoulli process offers a more advanced treatment. Since a Bernoulli trial has only two possible outcomes, it can be framed as some "yes or no" question. For example: *Is the top card of a shuffled deck an ace? *Was the newborn child a girl? (See human sex ratio.) Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally. The term "success" in this sense consists in the result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the random variable and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas ''beta distribution of the second kind'' is an alternative name for the beta prime distribution. The generalization to mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
An Essay Towards Solving A Problem In The Doctrine Of Chances
''An Essay towards solving a Problem in the Doctrine of Chances'' is a work on the mathematical theory of probability by Thomas Bayes, published in 1763, two years after its author's death, and containing multiple amendments and additions due to his friend Richard Price. The title comes from the contemporary use of the phrase "doctrine of chances" to mean the theory of probability, which had been introduced via the title of a book by Abraham de Moivre. Contemporary reprints of the Essay carry a more specific and significant title: ''A Method of Calculating the Exact Probability of All Conclusions founded on Induction''. The essay includes theorems of conditional probability which form the basis of what is now called Bayes's Theorem, together with a detailed treatment of the problem of setting a prior probability. Bayes supposed a sequence of independent experiments, each having as its outcome either success or failure, the probability of success being some number ''p'' between ... [...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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical consequences to the outcome. There are three branches of decision theory: # Normative decision theory: Concerned with the identification of optimal decisions, where optimality is often determined by considering an ideal decision-maker who is able to calculate with perfect accuracy and is in some sense fully rational. # Prescriptive decision theory: Concerned with describing observed behaviors through the use of conceptual models, under the assumption that those making the decisions are behaving under some consistent rules. # Descriptive decision theory: Analyzes how individuals actually make the decisions that they do. Decision theory is closely related to the field of game theory and is an interdisciplinary topic, studied by economi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dutch Book
In gambling, a Dutch book or lock is a set of odds and bets, established by the bookmaker, that ensures that the bookmaker will profit—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which the gamblers bet. It is associated with probabilities implied by the odds not being coherent. In economics, the term usually refers to a sequence of trades that would leave one party strictly worse off and another strictly better off. Typical assumptions in consumer choice theory rule out the possibility that anyone can be Dutch-booked. In philosophy it is used to explore degrees of certainty of beliefs. There is no agreement on the etymology of the term.Hajek, Alan ''Dutch Book Arguments'', Chapter 7 in The Oxford Handbook of Rational and Social Choice, ed. Paul Anand, Prasanta Pattanaik, and Clemens Puppe, 173-195, Oxford University Press Gambling The main point of the Dutch book argument is to show that rational people must hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cox's Theorem
Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability, as the laws of probability derived by Cox's theorem are applicable to any proposition. Logical (also known as objective Bayesian) probability is a type of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications. Cox's assumptions Cox wanted his system to satisfy the following conditions: #Divisibility and comparability – The plausibility of a proposition is a real number and is dependent on information we have related to the proposition. #Common sense – Plausibilities should vary sensibly with the assessment of plausibilities in the model. #Consistency – If the plausibility of a proposition can be derived in many ways, all the results must be equal. The post ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
